src/HOL/GCD.thy
author nipkow
Tue Jun 23 12:58:53 2009 +0200 (2009-06-23)
changeset 31766 f767c5b1702e
parent 31734 a4a79836d07b
child 31798 fe9a3043d36c
permissions -rw-r--r--
new lemmas
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(*  Title:      GCD.thy
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    Authors:    Christophe Tabacznyj, Lawrence C. Paulson, Amine Chaieb,
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                Thomas M. Rasmussen, Jeremy Avigad
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This file deals with the functions gcd and lcm, and properties of
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primes. Definitions and lemmas are proved uniformly for the natural
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numbers and integers.
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This file combines and revises a number of prior developments.
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The original theories "GCD" and "Primes" were by Christophe Tabacznyj
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and Lawrence C. Paulson, based on \cite{davenport92}. They introduced
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gcd, lcm, and prime for the natural numbers.
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The original theory "IntPrimes" was by Thomas M. Rasmussen, and
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extended gcd, lcm, primes to the integers. Amine Chaieb provided
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another extension of the notions to the integers, and added a number
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of results to "Primes" and "GCD". IntPrimes also defined and developed
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the congruence relations on the integers. The notion was extended to
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the natural numbers by Chiaeb.
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*)
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header {* GCD *}
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theory GCD
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imports NatTransfer
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begin
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declare One_nat_def [simp del]
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subsection {* gcd *}
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class gcd = one +
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fixes
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  gcd :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" and
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  lcm :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"
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begin
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abbreviation
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  coprime :: "'a \<Rightarrow> 'a \<Rightarrow> bool"
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where
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  "coprime x y == (gcd x y = 1)"
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end
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class prime = one +
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fixes
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  prime :: "'a \<Rightarrow> bool"
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(* definitions for the natural numbers *)
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instantiation nat :: gcd
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begin
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fun
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  gcd_nat  :: "nat \<Rightarrow> nat \<Rightarrow> nat"
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where
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  "gcd_nat x y =
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   (if y = 0 then x else gcd y (x mod y))"
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definition
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  lcm_nat :: "nat \<Rightarrow> nat \<Rightarrow> nat"
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where
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  "lcm_nat x y = x * y div (gcd x y)"
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instance proof qed
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end
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instantiation nat :: prime
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begin
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definition
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  prime_nat :: "nat \<Rightarrow> bool"
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where
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  [code del]: "prime_nat p = (1 < p \<and> (\<forall>m. m dvd p --> m = 1 \<or> m = p))"
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instance proof qed
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end
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(* definitions for the integers *)
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instantiation int :: gcd
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begin
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definition
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  gcd_int  :: "int \<Rightarrow> int \<Rightarrow> int"
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where
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  "gcd_int x y = int (gcd (nat (abs x)) (nat (abs y)))"
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definition
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  lcm_int :: "int \<Rightarrow> int \<Rightarrow> int"
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where
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  "lcm_int x y = int (lcm (nat (abs x)) (nat (abs y)))"
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instance proof qed
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end
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instantiation int :: prime
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begin
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definition
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  prime_int :: "int \<Rightarrow> bool"
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where
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  [code del]: "prime_int p = prime (nat p)"
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instance proof qed
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end
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subsection {* Set up Transfer *}
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lemma transfer_nat_int_gcd:
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  "(x::int) >= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> gcd (nat x) (nat y) = nat (gcd x y)"
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  "(x::int) >= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> lcm (nat x) (nat y) = nat (lcm x y)"
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  "(x::int) >= 0 \<Longrightarrow> prime (nat x) = prime x"
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  unfolding gcd_int_def lcm_int_def prime_int_def
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  by auto
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lemma transfer_nat_int_gcd_closures:
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  "x >= (0::int) \<Longrightarrow> y >= 0 \<Longrightarrow> gcd x y >= 0"
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  "x >= (0::int) \<Longrightarrow> y >= 0 \<Longrightarrow> lcm x y >= 0"
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  by (auto simp add: gcd_int_def lcm_int_def)
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declare TransferMorphism_nat_int[transfer add return:
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    transfer_nat_int_gcd transfer_nat_int_gcd_closures]
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lemma transfer_int_nat_gcd:
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  "gcd (int x) (int y) = int (gcd x y)"
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  "lcm (int x) (int y) = int (lcm x y)"
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  "prime (int x) = prime x"
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  by (unfold gcd_int_def lcm_int_def prime_int_def, auto)
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lemma transfer_int_nat_gcd_closures:
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  "is_nat x \<Longrightarrow> is_nat y \<Longrightarrow> gcd x y >= 0"
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  "is_nat x \<Longrightarrow> is_nat y \<Longrightarrow> lcm x y >= 0"
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  by (auto simp add: gcd_int_def lcm_int_def)
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declare TransferMorphism_int_nat[transfer add return:
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    transfer_int_nat_gcd transfer_int_nat_gcd_closures]
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subsection {* GCD *}
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(* was gcd_induct *)
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lemma nat_gcd_induct:
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  fixes m n :: nat
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  assumes "\<And>m. P m 0"
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    and "\<And>m n. 0 < n \<Longrightarrow> P n (m mod n) \<Longrightarrow> P m n"
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  shows "P m n"
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  apply (rule gcd_nat.induct)
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  apply (case_tac "y = 0")
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  using assms apply simp_all
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done
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(* specific to int *)
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lemma int_gcd_neg1 [simp]: "gcd (-x::int) y = gcd x y"
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  by (simp add: gcd_int_def)
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lemma int_gcd_neg2 [simp]: "gcd (x::int) (-y) = gcd x y"
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  by (simp add: gcd_int_def)
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lemma int_gcd_abs: "gcd (x::int) y = gcd (abs x) (abs y)"
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  by (simp add: gcd_int_def)
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lemma int_gcd_cases:
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  fixes x :: int and y
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  assumes "x >= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> P (gcd x y)"
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      and "x >= 0 \<Longrightarrow> y <= 0 \<Longrightarrow> P (gcd x (-y))"
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      and "x <= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> P (gcd (-x) y)"
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      and "x <= 0 \<Longrightarrow> y <= 0 \<Longrightarrow> P (gcd (-x) (-y))"
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  shows "P (gcd x y)"
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by (insert prems, auto simp add: int_gcd_neg1 int_gcd_neg2, arith)
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lemma int_gcd_ge_0 [simp]: "gcd (x::int) y >= 0"
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  by (simp add: gcd_int_def)
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lemma int_lcm_neg1: "lcm (-x::int) y = lcm x y"
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  by (simp add: lcm_int_def)
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lemma int_lcm_neg2: "lcm (x::int) (-y) = lcm x y"
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  by (simp add: lcm_int_def)
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lemma int_lcm_abs: "lcm (x::int) y = lcm (abs x) (abs y)"
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  by (simp add: lcm_int_def)
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lemma int_lcm_cases:
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  fixes x :: int and y
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  assumes "x >= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> P (lcm x y)"
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      and "x >= 0 \<Longrightarrow> y <= 0 \<Longrightarrow> P (lcm x (-y))"
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      and "x <= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> P (lcm (-x) y)"
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      and "x <= 0 \<Longrightarrow> y <= 0 \<Longrightarrow> P (lcm (-x) (-y))"
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  shows "P (lcm x y)"
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by (insert prems, auto simp add: int_lcm_neg1 int_lcm_neg2, arith)
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lemma int_lcm_ge_0 [simp]: "lcm (x::int) y >= 0"
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  by (simp add: lcm_int_def)
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(* was gcd_0, etc. *)
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lemma nat_gcd_0 [simp]: "gcd (x::nat) 0 = x"
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  by simp
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(* was igcd_0, etc. *)
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lemma int_gcd_0 [simp]: "gcd (x::int) 0 = abs x"
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  by (unfold gcd_int_def, auto)
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lemma nat_gcd_0_left [simp]: "gcd 0 (x::nat) = x"
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  by simp
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lemma int_gcd_0_left [simp]: "gcd 0 (x::int) = abs x"
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  by (unfold gcd_int_def, auto)
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lemma nat_gcd_red: "gcd (x::nat) y = gcd y (x mod y)"
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  by (case_tac "y = 0", auto)
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(* weaker, but useful for the simplifier *)
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lemma nat_gcd_non_0: "y ~= (0::nat) \<Longrightarrow> gcd (x::nat) y = gcd y (x mod y)"
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  by simp
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lemma nat_gcd_1 [simp]: "gcd (m::nat) 1 = 1"
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  by simp
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lemma nat_gcd_Suc_0 [simp]: "gcd (m::nat) (Suc 0) = Suc 0"
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  by (simp add: One_nat_def)
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lemma int_gcd_1 [simp]: "gcd (m::int) 1 = 1"
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  by (simp add: gcd_int_def)
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lemma nat_gcd_self [simp]: "gcd (x::nat) x = x"
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  by simp
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lemma int_gcd_def [simp]: "gcd (x::int) x = abs x"
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  by (subst int_gcd_abs, auto simp add: gcd_int_def)
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declare gcd_nat.simps [simp del]
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text {*
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  \medskip @{term "gcd m n"} divides @{text m} and @{text n}.  The
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  conjunctions don't seem provable separately.
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*}
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lemma nat_gcd_dvd1 [iff]: "(gcd (m::nat)) n dvd m"
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  and nat_gcd_dvd2 [iff]: "(gcd m n) dvd n"
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  apply (induct m n rule: nat_gcd_induct)
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  apply (simp_all add: nat_gcd_non_0)
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  apply (blast dest: dvd_mod_imp_dvd)
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done
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thm nat_gcd_dvd1 [transferred]
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lemma int_gcd_dvd1 [iff]: "gcd (x::int) y dvd x"
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  apply (subst int_gcd_abs)
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  apply (rule dvd_trans)
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  apply (rule nat_gcd_dvd1 [transferred])
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  apply auto
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done
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lemma int_gcd_dvd2 [iff]: "gcd (x::int) y dvd y"
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  apply (subst int_gcd_abs)
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  apply (rule dvd_trans)
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  apply (rule nat_gcd_dvd2 [transferred])
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  apply auto
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done
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lemma dvd_gcd_D1_nat: "k dvd gcd m n \<Longrightarrow> (k::nat) dvd m"
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by(metis nat_gcd_dvd1 dvd_trans)
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lemma dvd_gcd_D2_nat: "k dvd gcd m n \<Longrightarrow> (k::nat) dvd n"
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by(metis nat_gcd_dvd2 dvd_trans)
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lemma dvd_gcd_D1_int: "i dvd gcd m n \<Longrightarrow> (i::int) dvd m"
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by(metis int_gcd_dvd1 dvd_trans)
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lemma dvd_gcd_D2_int: "i dvd gcd m n \<Longrightarrow> (i::int) dvd n"
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by(metis int_gcd_dvd2 dvd_trans)
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lemma nat_gcd_le1 [simp]: "a \<noteq> 0 \<Longrightarrow> gcd (a::nat) b \<le> a"
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  by (rule dvd_imp_le, auto)
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lemma nat_gcd_le2 [simp]: "b \<noteq> 0 \<Longrightarrow> gcd (a::nat) b \<le> b"
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  by (rule dvd_imp_le, auto)
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lemma int_gcd_le1 [simp]: "a > 0 \<Longrightarrow> gcd (a::int) b \<le> a"
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  by (rule zdvd_imp_le, auto)
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lemma int_gcd_le2 [simp]: "b > 0 \<Longrightarrow> gcd (a::int) b \<le> b"
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  by (rule zdvd_imp_le, auto)
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lemma nat_gcd_greatest: "(k::nat) dvd m \<Longrightarrow> k dvd n \<Longrightarrow> k dvd gcd m n"
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  by (induct m n rule: nat_gcd_induct) (simp_all add: nat_gcd_non_0 dvd_mod)
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lemma int_gcd_greatest:
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  assumes "(k::int) dvd m" and "k dvd n"
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  shows "k dvd gcd m n"
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  apply (subst int_gcd_abs)
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  apply (subst abs_dvd_iff [symmetric])
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  apply (rule nat_gcd_greatest [transferred])
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  using prems apply auto
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done
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lemma nat_gcd_greatest_iff [iff]: "(k dvd gcd (m::nat) n) =
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    (k dvd m & k dvd n)"
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  by (blast intro!: nat_gcd_greatest intro: dvd_trans)
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lemma int_gcd_greatest_iff: "((k::int) dvd gcd m n) = (k dvd m & k dvd n)"
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  by (blast intro!: int_gcd_greatest intro: dvd_trans)
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lemma nat_gcd_zero [simp]: "(gcd (m::nat) n = 0) = (m = 0 & n = 0)"
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  by (simp only: dvd_0_left_iff [symmetric] nat_gcd_greatest_iff)
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lemma int_gcd_zero [simp]: "(gcd (m::int) n = 0) = (m = 0 & n = 0)"
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  by (auto simp add: gcd_int_def)
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lemma nat_gcd_pos [simp]: "(gcd (m::nat) n > 0) = (m ~= 0 | n ~= 0)"
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  by (insert nat_gcd_zero [of m n], arith)
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lemma int_gcd_pos [simp]: "(gcd (m::int) n > 0) = (m ~= 0 | n ~= 0)"
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  by (insert int_gcd_zero [of m n], insert int_gcd_ge_0 [of m n], arith)
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   341
huffman@31706
   342
lemma nat_gcd_commute: "gcd (m::nat) n = gcd n m"
huffman@31706
   343
  by (rule dvd_anti_sym, auto)
haftmann@23687
   344
huffman@31706
   345
lemma int_gcd_commute: "gcd (m::int) n = gcd n m"
huffman@31706
   346
  by (auto simp add: gcd_int_def nat_gcd_commute)
huffman@31706
   347
huffman@31706
   348
lemma nat_gcd_assoc: "gcd (gcd (k::nat) m) n = gcd k (gcd m n)"
huffman@31706
   349
  apply (rule dvd_anti_sym)
huffman@31706
   350
  apply (blast intro: dvd_trans)+
huffman@31706
   351
done
wenzelm@21256
   352
huffman@31706
   353
lemma int_gcd_assoc: "gcd (gcd (k::int) m) n = gcd k (gcd m n)"
huffman@31706
   354
  by (auto simp add: gcd_int_def nat_gcd_assoc)
huffman@31706
   355
nipkow@31766
   356
lemmas nat_gcd_left_commute =
nipkow@31766
   357
  mk_left_commute[of gcd, OF nat_gcd_assoc nat_gcd_commute]
huffman@31706
   358
nipkow@31766
   359
lemmas int_gcd_left_commute =
nipkow@31766
   360
  mk_left_commute[of gcd, OF int_gcd_assoc int_gcd_commute]
huffman@31706
   361
huffman@31706
   362
lemmas nat_gcd_ac = nat_gcd_assoc nat_gcd_commute nat_gcd_left_commute
huffman@31706
   363
  -- {* gcd is an AC-operator *}
wenzelm@21256
   364
huffman@31706
   365
lemmas int_gcd_ac = int_gcd_assoc int_gcd_commute int_gcd_left_commute
huffman@31706
   366
huffman@31706
   367
lemma nat_gcd_1_left [simp]: "gcd (1::nat) m = 1"
huffman@31706
   368
  by (subst nat_gcd_commute, simp)
huffman@31706
   369
huffman@31706
   370
lemma nat_gcd_Suc_0_left [simp]: "gcd (Suc 0) m = Suc 0"
huffman@31706
   371
  by (subst nat_gcd_commute, simp add: One_nat_def)
huffman@31706
   372
huffman@31706
   373
lemma int_gcd_1_left [simp]: "gcd (1::int) m = 1"
huffman@31706
   374
  by (subst int_gcd_commute, simp)
wenzelm@21256
   375
huffman@31706
   376
lemma nat_gcd_unique: "(d::nat) dvd a \<and> d dvd b \<and>
huffman@31706
   377
    (\<forall>e. e dvd a \<and> e dvd b \<longrightarrow> e dvd d) \<longleftrightarrow> d = gcd a b"
huffman@31706
   378
  apply auto
huffman@31706
   379
  apply (rule dvd_anti_sym)
huffman@31706
   380
  apply (erule (1) nat_gcd_greatest)
huffman@31706
   381
  apply auto
huffman@31706
   382
done
wenzelm@21256
   383
huffman@31706
   384
lemma int_gcd_unique: "d >= 0 & (d::int) dvd a \<and> d dvd b \<and>
huffman@31706
   385
    (\<forall>e. e dvd a \<and> e dvd b \<longrightarrow> e dvd d) \<longleftrightarrow> d = gcd a b"
huffman@31706
   386
  apply (case_tac "d = 0")
huffman@31706
   387
  apply force
huffman@31706
   388
  apply (rule iffI)
huffman@31706
   389
  apply (rule zdvd_anti_sym)
huffman@31706
   390
  apply arith
huffman@31706
   391
  apply (subst int_gcd_pos)
huffman@31706
   392
  apply clarsimp
huffman@31706
   393
  apply (drule_tac x = "d + 1" in spec)
huffman@31706
   394
  apply (frule zdvd_imp_le)
huffman@31706
   395
  apply (auto intro: int_gcd_greatest)
huffman@31706
   396
done
huffman@30082
   397
wenzelm@21256
   398
text {*
wenzelm@21256
   399
  \medskip Multiplication laws
wenzelm@21256
   400
*}
wenzelm@21256
   401
huffman@31706
   402
lemma nat_gcd_mult_distrib: "(k::nat) * gcd m n = gcd (k * m) (k * n)"
wenzelm@21256
   403
    -- {* \cite[page 27]{davenport92} *}
huffman@31706
   404
  apply (induct m n rule: nat_gcd_induct)
huffman@31706
   405
  apply simp
wenzelm@21256
   406
  apply (case_tac "k = 0")
huffman@31706
   407
  apply (simp_all add: mod_geq nat_gcd_non_0 mod_mult_distrib2)
huffman@31706
   408
done
wenzelm@21256
   409
huffman@31706
   410
lemma int_gcd_mult_distrib: "abs (k::int) * gcd m n = gcd (k * m) (k * n)"
huffman@31706
   411
  apply (subst (1 2) int_gcd_abs)
huffman@31706
   412
  apply (simp add: abs_mult)
huffman@31706
   413
  apply (rule nat_gcd_mult_distrib [transferred])
huffman@31706
   414
  apply auto
huffman@31706
   415
done
wenzelm@21256
   416
huffman@31706
   417
lemma nat_gcd_mult [simp]: "gcd (k::nat) (k * n) = k"
huffman@31706
   418
  by (rule nat_gcd_mult_distrib [of k 1 n, simplified, symmetric])
wenzelm@21256
   419
huffman@31706
   420
lemma int_gcd_mult [simp]: "gcd (k::int) (k * n) = abs k"
huffman@31706
   421
  by (rule int_gcd_mult_distrib [of k 1 n, simplified, symmetric])
huffman@31706
   422
huffman@31706
   423
lemma nat_coprime_dvd_mult: "coprime (k::nat) n \<Longrightarrow> k dvd m * n \<Longrightarrow> k dvd m"
huffman@31706
   424
  apply (insert nat_gcd_mult_distrib [of m k n])
wenzelm@21256
   425
  apply simp
wenzelm@21256
   426
  apply (erule_tac t = m in ssubst)
wenzelm@21256
   427
  apply simp
wenzelm@21256
   428
  done
wenzelm@21256
   429
huffman@31706
   430
lemma int_coprime_dvd_mult:
huffman@31706
   431
  assumes "coprime (k::int) n" and "k dvd m * n"
huffman@31706
   432
  shows "k dvd m"
wenzelm@21256
   433
huffman@31706
   434
  using prems
huffman@31706
   435
  apply (subst abs_dvd_iff [symmetric])
huffman@31706
   436
  apply (subst dvd_abs_iff [symmetric])
huffman@31706
   437
  apply (subst (asm) int_gcd_abs)
huffman@31706
   438
  apply (rule nat_coprime_dvd_mult [transferred])
huffman@31706
   439
  apply auto
huffman@31706
   440
  apply (subst abs_mult [symmetric], auto)
huffman@31706
   441
done
huffman@31706
   442
huffman@31706
   443
lemma nat_coprime_dvd_mult_iff: "coprime (k::nat) n \<Longrightarrow>
huffman@31706
   444
    (k dvd m * n) = (k dvd m)"
huffman@31706
   445
  by (auto intro: nat_coprime_dvd_mult)
huffman@31706
   446
huffman@31706
   447
lemma int_coprime_dvd_mult_iff: "coprime (k::int) n \<Longrightarrow>
huffman@31706
   448
    (k dvd m * n) = (k dvd m)"
huffman@31706
   449
  by (auto intro: int_coprime_dvd_mult)
huffman@31706
   450
huffman@31706
   451
lemma nat_gcd_mult_cancel: "coprime k n \<Longrightarrow> gcd ((k::nat) * m) n = gcd m n"
wenzelm@21256
   452
  apply (rule dvd_anti_sym)
huffman@31706
   453
  apply (rule nat_gcd_greatest)
huffman@31706
   454
  apply (rule_tac n = k in nat_coprime_dvd_mult)
huffman@31706
   455
  apply (simp add: nat_gcd_assoc)
huffman@31706
   456
  apply (simp add: nat_gcd_commute)
huffman@31706
   457
  apply (simp_all add: mult_commute)
huffman@31706
   458
done
wenzelm@21256
   459
huffman@31706
   460
lemma int_gcd_mult_cancel:
huffman@31706
   461
  assumes "coprime (k::int) n"
huffman@31706
   462
  shows "gcd (k * m) n = gcd m n"
huffman@31706
   463
huffman@31706
   464
  using prems
huffman@31706
   465
  apply (subst (1 2) int_gcd_abs)
huffman@31706
   466
  apply (subst abs_mult)
huffman@31706
   467
  apply (rule nat_gcd_mult_cancel [transferred])
huffman@31706
   468
  apply (auto simp add: int_gcd_abs [symmetric])
huffman@31706
   469
done
wenzelm@21256
   470
wenzelm@21256
   471
text {* \medskip Addition laws *}
wenzelm@21256
   472
huffman@31706
   473
lemma nat_gcd_add1 [simp]: "gcd ((m::nat) + n) n = gcd m n"
huffman@31706
   474
  apply (case_tac "n = 0")
huffman@31706
   475
  apply (simp_all add: nat_gcd_non_0)
huffman@31706
   476
done
huffman@31706
   477
huffman@31706
   478
lemma nat_gcd_add2 [simp]: "gcd (m::nat) (m + n) = gcd m n"
huffman@31706
   479
  apply (subst (1 2) nat_gcd_commute)
huffman@31706
   480
  apply (subst add_commute)
huffman@31706
   481
  apply simp
huffman@31706
   482
done
huffman@31706
   483
huffman@31706
   484
(* to do: add the other variations? *)
huffman@31706
   485
huffman@31706
   486
lemma nat_gcd_diff1: "(m::nat) >= n \<Longrightarrow> gcd (m - n) n = gcd m n"
huffman@31706
   487
  by (subst nat_gcd_add1 [symmetric], auto)
huffman@31706
   488
huffman@31706
   489
lemma nat_gcd_diff2: "(n::nat) >= m \<Longrightarrow> gcd (n - m) n = gcd m n"
huffman@31706
   490
  apply (subst nat_gcd_commute)
huffman@31706
   491
  apply (subst nat_gcd_diff1 [symmetric])
huffman@31706
   492
  apply auto
huffman@31706
   493
  apply (subst nat_gcd_commute)
huffman@31706
   494
  apply (subst nat_gcd_diff1)
huffman@31706
   495
  apply assumption
huffman@31706
   496
  apply (rule nat_gcd_commute)
huffman@31706
   497
done
huffman@31706
   498
huffman@31706
   499
lemma int_gcd_non_0: "(y::int) > 0 \<Longrightarrow> gcd x y = gcd y (x mod y)"
huffman@31706
   500
  apply (frule_tac b = y and a = x in pos_mod_sign)
huffman@31706
   501
  apply (simp del: pos_mod_sign add: gcd_int_def abs_if nat_mod_distrib)
huffman@31706
   502
  apply (auto simp add: nat_gcd_non_0 nat_mod_distrib [symmetric]
huffman@31706
   503
    zmod_zminus1_eq_if)
huffman@31706
   504
  apply (frule_tac a = x in pos_mod_bound)
huffman@31706
   505
  apply (subst (1 2) nat_gcd_commute)
huffman@31706
   506
  apply (simp del: pos_mod_bound add: nat_diff_distrib nat_gcd_diff2
huffman@31706
   507
    nat_le_eq_zle)
huffman@31706
   508
done
wenzelm@21256
   509
huffman@31706
   510
lemma int_gcd_red: "gcd (x::int) y = gcd y (x mod y)"
huffman@31706
   511
  apply (case_tac "y = 0")
huffman@31706
   512
  apply force
huffman@31706
   513
  apply (case_tac "y > 0")
huffman@31706
   514
  apply (subst int_gcd_non_0, auto)
huffman@31706
   515
  apply (insert int_gcd_non_0 [of "-y" "-x"])
huffman@31706
   516
  apply (auto simp add: int_gcd_neg1 int_gcd_neg2)
huffman@31706
   517
done
huffman@31706
   518
huffman@31706
   519
lemma int_gcd_add1 [simp]: "gcd ((m::int) + n) n = gcd m n"
huffman@31706
   520
  apply (case_tac "n = 0", force)
huffman@31706
   521
  apply (subst (1 2) int_gcd_red)
huffman@31706
   522
  apply auto
huffman@31706
   523
done
huffman@31706
   524
huffman@31706
   525
lemma int_gcd_add2 [simp]: "gcd m ((m::int) + n) = gcd m n"
huffman@31706
   526
  apply (subst int_gcd_commute)
huffman@31706
   527
  apply (subst add_commute)
huffman@31706
   528
  apply (subst int_gcd_add1)
huffman@31706
   529
  apply (subst int_gcd_commute)
huffman@31706
   530
  apply (rule refl)
huffman@31706
   531
done
wenzelm@21256
   532
huffman@31706
   533
lemma nat_gcd_add_mult: "gcd (m::nat) (k * m + n) = gcd m n"
huffman@31706
   534
  by (induct k, simp_all add: ring_simps)
wenzelm@21256
   535
huffman@31706
   536
lemma int_gcd_add_mult: "gcd (m::int) (k * m + n) = gcd m n"
huffman@31706
   537
  apply (subgoal_tac "ALL s. ALL m. ALL n.
huffman@31706
   538
      gcd m (int (s::nat) * m + n) = gcd m n")
huffman@31706
   539
  apply (case_tac "k >= 0")
huffman@31706
   540
  apply (drule_tac x = "nat k" in spec, force)
huffman@31706
   541
  apply (subst (1 2) int_gcd_neg2 [symmetric])
huffman@31706
   542
  apply (drule_tac x = "nat (- k)" in spec)
huffman@31706
   543
  apply (drule_tac x = "m" in spec)
huffman@31706
   544
  apply (drule_tac x = "-n" in spec)
huffman@31706
   545
  apply auto
huffman@31706
   546
  apply (rule nat_induct)
huffman@31706
   547
  apply auto
huffman@31706
   548
  apply (auto simp add: left_distrib)
huffman@31706
   549
  apply (subst add_assoc)
huffman@31706
   550
  apply simp
huffman@31706
   551
done
wenzelm@21256
   552
huffman@31706
   553
(* to do: differences, and all variations of addition rules
huffman@31706
   554
    as simplification rules for nat and int *)
huffman@31706
   555
huffman@31706
   556
lemma nat_gcd_dvd_prod [iff]: "gcd (m::nat) n dvd k * n"
haftmann@23687
   557
  using mult_dvd_mono [of 1] by auto
chaieb@22027
   558
huffman@31706
   559
(* to do: add the three variations of these, and for ints? *)
huffman@31706
   560
nipkow@31734
   561
lemma finite_divisors_nat:
nipkow@31734
   562
  assumes "(m::nat)~= 0" shows "finite{d. d dvd m}"
nipkow@31734
   563
proof-
nipkow@31734
   564
  have "finite{d. d <= m}" by(blast intro: bounded_nat_set_is_finite)
nipkow@31734
   565
  from finite_subset[OF _ this] show ?thesis using assms
nipkow@31734
   566
    by(bestsimp intro!:dvd_imp_le)
nipkow@31734
   567
qed
nipkow@31734
   568
nipkow@31734
   569
lemma finite_divisors_int:
nipkow@31734
   570
  assumes "(i::int) ~= 0" shows "finite{d. d dvd i}"
nipkow@31734
   571
proof-
nipkow@31734
   572
  have "{d. abs d <= abs i} = {- abs i .. abs i}" by(auto simp:abs_if)
nipkow@31734
   573
  hence "finite{d. abs d <= abs i}" by simp
nipkow@31734
   574
  from finite_subset[OF _ this] show ?thesis using assms
nipkow@31734
   575
    by(bestsimp intro!:dvd_imp_le_int)
nipkow@31734
   576
qed
nipkow@31734
   577
nipkow@31734
   578
lemma gcd_is_Max_divisors_nat:
nipkow@31734
   579
  "m ~= 0 \<Longrightarrow> n ~= 0 \<Longrightarrow> gcd (m::nat) n = (Max {d. d dvd m & d dvd n})"
nipkow@31734
   580
apply(rule Max_eqI[THEN sym])
nipkow@31734
   581
  apply (metis dvd.eq_iff finite_Collect_conjI finite_divisors_nat)
nipkow@31734
   582
 apply simp
nipkow@31734
   583
 apply(metis Suc_diff_1 Suc_neq_Zero dvd_imp_le nat_gcd_greatest_iff nat_gcd_pos)
nipkow@31734
   584
apply simp
nipkow@31734
   585
done
nipkow@31734
   586
nipkow@31734
   587
lemma gcd_is_Max_divisors_int:
nipkow@31734
   588
  "m ~= 0 ==> n ~= 0 ==> gcd (m::int) n = (Max {d. d dvd m & d dvd n})"
nipkow@31734
   589
apply(rule Max_eqI[THEN sym])
nipkow@31734
   590
  apply (metis dvd.eq_iff finite_Collect_conjI finite_divisors_int)
nipkow@31734
   591
 apply simp
nipkow@31734
   592
 apply (metis int_gcd_greatest_iff int_gcd_pos zdvd_imp_le)
nipkow@31734
   593
apply simp
nipkow@31734
   594
done
nipkow@31734
   595
chaieb@22027
   596
huffman@31706
   597
subsection {* Coprimality *}
huffman@31706
   598
huffman@31706
   599
lemma nat_div_gcd_coprime [intro]:
huffman@31706
   600
  assumes nz: "(a::nat) \<noteq> 0 \<or> b \<noteq> 0"
huffman@31706
   601
  shows "coprime (a div gcd a b) (b div gcd a b)"
wenzelm@22367
   602
proof -
haftmann@27556
   603
  let ?g = "gcd a b"
chaieb@22027
   604
  let ?a' = "a div ?g"
chaieb@22027
   605
  let ?b' = "b div ?g"
haftmann@27556
   606
  let ?g' = "gcd ?a' ?b'"
chaieb@22027
   607
  have dvdg: "?g dvd a" "?g dvd b" by simp_all
chaieb@22027
   608
  have dvdg': "?g' dvd ?a'" "?g' dvd ?b'" by simp_all
wenzelm@22367
   609
  from dvdg dvdg' obtain ka kb ka' kb' where
wenzelm@22367
   610
      kab: "a = ?g * ka" "b = ?g * kb" "?a' = ?g' * ka'" "?b' = ?g' * kb'"
chaieb@22027
   611
    unfolding dvd_def by blast
huffman@31706
   612
  then have "?g * ?a' = (?g * ?g') * ka'" "?g * ?b' = (?g * ?g') * kb'"
huffman@31706
   613
    by simp_all
wenzelm@22367
   614
  then have dvdgg':"?g * ?g' dvd a" "?g* ?g' dvd b"
wenzelm@22367
   615
    by (auto simp add: dvd_mult_div_cancel [OF dvdg(1)]
wenzelm@22367
   616
      dvd_mult_div_cancel [OF dvdg(2)] dvd_def)
huffman@31706
   617
  have "?g \<noteq> 0" using nz by (simp add: nat_gcd_zero)
huffman@31706
   618
  then have gp: "?g > 0" by arith
huffman@31706
   619
  from nat_gcd_greatest [OF dvdgg'] have "?g * ?g' dvd ?g" .
wenzelm@22367
   620
  with dvd_mult_cancel1 [OF gp] show "?g' = 1" by simp
chaieb@22027
   621
qed
chaieb@22027
   622
huffman@31706
   623
lemma int_div_gcd_coprime [intro]:
huffman@31706
   624
  assumes nz: "(a::int) \<noteq> 0 \<or> b \<noteq> 0"
huffman@31706
   625
  shows "coprime (a div gcd a b) (b div gcd a b)"
chaieb@27669
   626
huffman@31706
   627
  apply (subst (1 2 3) int_gcd_abs)
huffman@31706
   628
  apply (subst (1 2) abs_div)
huffman@31706
   629
  apply auto
huffman@31706
   630
  prefer 3
huffman@31706
   631
  apply (rule nat_div_gcd_coprime [transferred])
huffman@31706
   632
  using nz apply (auto simp add: int_gcd_abs [symmetric])
huffman@31706
   633
done
huffman@31706
   634
huffman@31706
   635
lemma nat_coprime: "coprime (a::nat) b \<longleftrightarrow> (\<forall>d. d dvd a \<and> d dvd b \<longleftrightarrow> d = 1)"
huffman@31706
   636
  using nat_gcd_unique[of 1 a b, simplified] by auto
huffman@31706
   637
huffman@31706
   638
lemma nat_coprime_Suc_0:
huffman@31706
   639
    "coprime (a::nat) b \<longleftrightarrow> (\<forall>d. d dvd a \<and> d dvd b \<longleftrightarrow> d = Suc 0)"
huffman@31706
   640
  using nat_coprime by (simp add: One_nat_def)
huffman@31706
   641
huffman@31706
   642
lemma int_coprime: "coprime (a::int) b \<longleftrightarrow>
huffman@31706
   643
    (\<forall>d. d >= 0 \<and> d dvd a \<and> d dvd b \<longleftrightarrow> d = 1)"
huffman@31706
   644
  using int_gcd_unique [of 1 a b]
huffman@31706
   645
  apply clarsimp
huffman@31706
   646
  apply (erule subst)
huffman@31706
   647
  apply (rule iffI)
huffman@31706
   648
  apply force
huffman@31706
   649
  apply (drule_tac x = "abs e" in exI)
huffman@31706
   650
  apply (case_tac "e >= 0")
huffman@31706
   651
  apply force
huffman@31706
   652
  apply force
huffman@31706
   653
done
huffman@31706
   654
huffman@31706
   655
lemma nat_gcd_coprime:
huffman@31706
   656
  assumes z: "gcd (a::nat) b \<noteq> 0" and a: "a = a' * gcd a b" and
huffman@31706
   657
    b: "b = b' * gcd a b"
huffman@31706
   658
  shows    "coprime a' b'"
huffman@31706
   659
huffman@31706
   660
  apply (subgoal_tac "a' = a div gcd a b")
huffman@31706
   661
  apply (erule ssubst)
huffman@31706
   662
  apply (subgoal_tac "b' = b div gcd a b")
huffman@31706
   663
  apply (erule ssubst)
huffman@31706
   664
  apply (rule nat_div_gcd_coprime)
huffman@31706
   665
  using prems
huffman@31706
   666
  apply force
huffman@31706
   667
  apply (subst (1) b)
huffman@31706
   668
  using z apply force
huffman@31706
   669
  apply (subst (1) a)
huffman@31706
   670
  using z apply force
huffman@31706
   671
done
huffman@31706
   672
huffman@31706
   673
lemma int_gcd_coprime:
huffman@31706
   674
  assumes z: "gcd (a::int) b \<noteq> 0" and a: "a = a' * gcd a b" and
huffman@31706
   675
    b: "b = b' * gcd a b"
huffman@31706
   676
  shows    "coprime a' b'"
huffman@31706
   677
huffman@31706
   678
  apply (subgoal_tac "a' = a div gcd a b")
huffman@31706
   679
  apply (erule ssubst)
huffman@31706
   680
  apply (subgoal_tac "b' = b div gcd a b")
huffman@31706
   681
  apply (erule ssubst)
huffman@31706
   682
  apply (rule int_div_gcd_coprime)
huffman@31706
   683
  using prems
huffman@31706
   684
  apply force
huffman@31706
   685
  apply (subst (1) b)
huffman@31706
   686
  using z apply force
huffman@31706
   687
  apply (subst (1) a)
huffman@31706
   688
  using z apply force
huffman@31706
   689
done
huffman@31706
   690
huffman@31706
   691
lemma nat_coprime_mult: assumes da: "coprime (d::nat) a" and db: "coprime d b"
huffman@31706
   692
    shows "coprime d (a * b)"
huffman@31706
   693
  apply (subst nat_gcd_commute)
huffman@31706
   694
  using da apply (subst nat_gcd_mult_cancel)
huffman@31706
   695
  apply (subst nat_gcd_commute, assumption)
huffman@31706
   696
  apply (subst nat_gcd_commute, rule db)
huffman@31706
   697
done
huffman@31706
   698
huffman@31706
   699
lemma int_coprime_mult: assumes da: "coprime (d::int) a" and db: "coprime d b"
huffman@31706
   700
    shows "coprime d (a * b)"
huffman@31706
   701
  apply (subst int_gcd_commute)
huffman@31706
   702
  using da apply (subst int_gcd_mult_cancel)
huffman@31706
   703
  apply (subst int_gcd_commute, assumption)
huffman@31706
   704
  apply (subst int_gcd_commute, rule db)
huffman@31706
   705
done
huffman@31706
   706
huffman@31706
   707
lemma nat_coprime_lmult:
huffman@31706
   708
  assumes dab: "coprime (d::nat) (a * b)" shows "coprime d a"
huffman@31706
   709
proof -
huffman@31706
   710
  have "gcd d a dvd gcd d (a * b)"
huffman@31706
   711
    by (rule nat_gcd_greatest, auto)
huffman@31706
   712
  with dab show ?thesis
huffman@31706
   713
    by auto
huffman@31706
   714
qed
huffman@31706
   715
huffman@31706
   716
lemma int_coprime_lmult:
huffman@31706
   717
  assumes dab: "coprime (d::int) (a * b)" shows "coprime d a"
huffman@31706
   718
proof -
huffman@31706
   719
  have "gcd d a dvd gcd d (a * b)"
huffman@31706
   720
    by (rule int_gcd_greatest, auto)
huffman@31706
   721
  with dab show ?thesis
huffman@31706
   722
    by auto
huffman@31706
   723
qed
huffman@31706
   724
huffman@31706
   725
lemma nat_coprime_rmult:
huffman@31706
   726
  assumes dab: "coprime (d::nat) (a * b)" shows "coprime d b"
huffman@31706
   727
proof -
huffman@31706
   728
  have "gcd d b dvd gcd d (a * b)"
huffman@31706
   729
    by (rule nat_gcd_greatest, auto intro: dvd_mult)
huffman@31706
   730
  with dab show ?thesis
huffman@31706
   731
    by auto
huffman@31706
   732
qed
huffman@31706
   733
huffman@31706
   734
lemma int_coprime_rmult:
huffman@31706
   735
  assumes dab: "coprime (d::int) (a * b)" shows "coprime d b"
huffman@31706
   736
proof -
huffman@31706
   737
  have "gcd d b dvd gcd d (a * b)"
huffman@31706
   738
    by (rule int_gcd_greatest, auto intro: dvd_mult)
huffman@31706
   739
  with dab show ?thesis
huffman@31706
   740
    by auto
huffman@31706
   741
qed
huffman@31706
   742
huffman@31706
   743
lemma nat_coprime_mul_eq: "coprime (d::nat) (a * b) \<longleftrightarrow>
huffman@31706
   744
    coprime d a \<and>  coprime d b"
huffman@31706
   745
  using nat_coprime_rmult[of d a b] nat_coprime_lmult[of d a b]
huffman@31706
   746
    nat_coprime_mult[of d a b]
huffman@31706
   747
  by blast
huffman@31706
   748
huffman@31706
   749
lemma int_coprime_mul_eq: "coprime (d::int) (a * b) \<longleftrightarrow>
huffman@31706
   750
    coprime d a \<and>  coprime d b"
huffman@31706
   751
  using int_coprime_rmult[of d a b] int_coprime_lmult[of d a b]
huffman@31706
   752
    int_coprime_mult[of d a b]
huffman@31706
   753
  by blast
huffman@31706
   754
huffman@31706
   755
lemma nat_gcd_coprime_exists:
huffman@31706
   756
    assumes nz: "gcd (a::nat) b \<noteq> 0"
huffman@31706
   757
    shows "\<exists>a' b'. a = a' * gcd a b \<and> b = b' * gcd a b \<and> coprime a' b'"
huffman@31706
   758
  apply (rule_tac x = "a div gcd a b" in exI)
huffman@31706
   759
  apply (rule_tac x = "b div gcd a b" in exI)
huffman@31706
   760
  using nz apply (auto simp add: nat_div_gcd_coprime dvd_div_mult)
huffman@31706
   761
done
huffman@31706
   762
huffman@31706
   763
lemma int_gcd_coprime_exists:
huffman@31706
   764
    assumes nz: "gcd (a::int) b \<noteq> 0"
huffman@31706
   765
    shows "\<exists>a' b'. a = a' * gcd a b \<and> b = b' * gcd a b \<and> coprime a' b'"
huffman@31706
   766
  apply (rule_tac x = "a div gcd a b" in exI)
huffman@31706
   767
  apply (rule_tac x = "b div gcd a b" in exI)
huffman@31706
   768
  using nz apply (auto simp add: int_div_gcd_coprime dvd_div_mult_self)
huffman@31706
   769
done
huffman@31706
   770
huffman@31706
   771
lemma nat_coprime_exp: "coprime (d::nat) a \<Longrightarrow> coprime d (a^n)"
huffman@31706
   772
  by (induct n, simp_all add: nat_coprime_mult)
huffman@31706
   773
huffman@31706
   774
lemma int_coprime_exp: "coprime (d::int) a \<Longrightarrow> coprime d (a^n)"
huffman@31706
   775
  by (induct n, simp_all add: int_coprime_mult)
huffman@31706
   776
huffman@31706
   777
lemma nat_coprime_exp2 [intro]: "coprime (a::nat) b \<Longrightarrow> coprime (a^n) (b^m)"
huffman@31706
   778
  apply (rule nat_coprime_exp)
huffman@31706
   779
  apply (subst nat_gcd_commute)
huffman@31706
   780
  apply (rule nat_coprime_exp)
huffman@31706
   781
  apply (subst nat_gcd_commute, assumption)
huffman@31706
   782
done
huffman@31706
   783
huffman@31706
   784
lemma int_coprime_exp2 [intro]: "coprime (a::int) b \<Longrightarrow> coprime (a^n) (b^m)"
huffman@31706
   785
  apply (rule int_coprime_exp)
huffman@31706
   786
  apply (subst int_gcd_commute)
huffman@31706
   787
  apply (rule int_coprime_exp)
huffman@31706
   788
  apply (subst int_gcd_commute, assumption)
huffman@31706
   789
done
huffman@31706
   790
huffman@31706
   791
lemma nat_gcd_exp: "gcd ((a::nat)^n) (b^n) = (gcd a b)^n"
huffman@31706
   792
proof (cases)
huffman@31706
   793
  assume "a = 0 & b = 0"
huffman@31706
   794
  thus ?thesis by simp
huffman@31706
   795
  next assume "~(a = 0 & b = 0)"
huffman@31706
   796
  hence "coprime ((a div gcd a b)^n) ((b div gcd a b)^n)"
huffman@31706
   797
    by auto
huffman@31706
   798
  hence "gcd ((a div gcd a b)^n * (gcd a b)^n)
huffman@31706
   799
      ((b div gcd a b)^n * (gcd a b)^n) = (gcd a b)^n"
huffman@31706
   800
    apply (subst (1 2) mult_commute)
huffman@31706
   801
    apply (subst nat_gcd_mult_distrib [symmetric])
huffman@31706
   802
    apply simp
huffman@31706
   803
    done
huffman@31706
   804
  also have "(a div gcd a b)^n * (gcd a b)^n = a^n"
huffman@31706
   805
    apply (subst div_power)
huffman@31706
   806
    apply auto
huffman@31706
   807
    apply (rule dvd_div_mult_self)
huffman@31706
   808
    apply (rule dvd_power_same)
huffman@31706
   809
    apply auto
huffman@31706
   810
    done
huffman@31706
   811
  also have "(b div gcd a b)^n * (gcd a b)^n = b^n"
huffman@31706
   812
    apply (subst div_power)
huffman@31706
   813
    apply auto
huffman@31706
   814
    apply (rule dvd_div_mult_self)
huffman@31706
   815
    apply (rule dvd_power_same)
huffman@31706
   816
    apply auto
huffman@31706
   817
    done
huffman@31706
   818
  finally show ?thesis .
huffman@31706
   819
qed
huffman@31706
   820
huffman@31706
   821
lemma int_gcd_exp: "gcd ((a::int)^n) (b^n) = (gcd a b)^n"
huffman@31706
   822
  apply (subst (1 2) int_gcd_abs)
huffman@31706
   823
  apply (subst (1 2) power_abs)
huffman@31706
   824
  apply (rule nat_gcd_exp [where n = n, transferred])
huffman@31706
   825
  apply auto
huffman@31706
   826
done
huffman@31706
   827
huffman@31706
   828
lemma nat_coprime_divprod: "(d::nat) dvd a * b  \<Longrightarrow> coprime d a \<Longrightarrow> d dvd b"
huffman@31706
   829
  using nat_coprime_dvd_mult_iff[of d a b]
huffman@31706
   830
  by (auto simp add: mult_commute)
huffman@31706
   831
huffman@31706
   832
lemma int_coprime_divprod: "(d::int) dvd a * b  \<Longrightarrow> coprime d a \<Longrightarrow> d dvd b"
huffman@31706
   833
  using int_coprime_dvd_mult_iff[of d a b]
huffman@31706
   834
  by (auto simp add: mult_commute)
huffman@31706
   835
huffman@31706
   836
lemma nat_division_decomp: assumes dc: "(a::nat) dvd b * c"
huffman@31706
   837
  shows "\<exists>b' c'. a = b' * c' \<and> b' dvd b \<and> c' dvd c"
huffman@31706
   838
proof-
huffman@31706
   839
  let ?g = "gcd a b"
huffman@31706
   840
  {assume "?g = 0" with dc have ?thesis by auto}
huffman@31706
   841
  moreover
huffman@31706
   842
  {assume z: "?g \<noteq> 0"
huffman@31706
   843
    from nat_gcd_coprime_exists[OF z]
huffman@31706
   844
    obtain a' b' where ab': "a = a' * ?g" "b = b' * ?g" "coprime a' b'"
huffman@31706
   845
      by blast
huffman@31706
   846
    have thb: "?g dvd b" by auto
huffman@31706
   847
    from ab'(1) have "a' dvd a"  unfolding dvd_def by blast
huffman@31706
   848
    with dc have th0: "a' dvd b*c" using dvd_trans[of a' a "b*c"] by simp
huffman@31706
   849
    from dc ab'(1,2) have "a'*?g dvd (b'*?g) *c" by auto
huffman@31706
   850
    hence "?g*a' dvd ?g * (b' * c)" by (simp add: mult_assoc)
huffman@31706
   851
    with z have th_1: "a' dvd b' * c" by auto
huffman@31706
   852
    from nat_coprime_dvd_mult[OF ab'(3)] th_1
huffman@31706
   853
    have thc: "a' dvd c" by (subst (asm) mult_commute, blast)
huffman@31706
   854
    from ab' have "a = ?g*a'" by algebra
huffman@31706
   855
    with thb thc have ?thesis by blast }
huffman@31706
   856
  ultimately show ?thesis by blast
huffman@31706
   857
qed
huffman@31706
   858
huffman@31706
   859
lemma int_division_decomp: assumes dc: "(a::int) dvd b * c"
huffman@31706
   860
  shows "\<exists>b' c'. a = b' * c' \<and> b' dvd b \<and> c' dvd c"
huffman@31706
   861
proof-
huffman@31706
   862
  let ?g = "gcd a b"
huffman@31706
   863
  {assume "?g = 0" with dc have ?thesis by auto}
huffman@31706
   864
  moreover
huffman@31706
   865
  {assume z: "?g \<noteq> 0"
huffman@31706
   866
    from int_gcd_coprime_exists[OF z]
huffman@31706
   867
    obtain a' b' where ab': "a = a' * ?g" "b = b' * ?g" "coprime a' b'"
huffman@31706
   868
      by blast
huffman@31706
   869
    have thb: "?g dvd b" by auto
huffman@31706
   870
    from ab'(1) have "a' dvd a"  unfolding dvd_def by blast
huffman@31706
   871
    with dc have th0: "a' dvd b*c"
huffman@31706
   872
      using dvd_trans[of a' a "b*c"] by simp
huffman@31706
   873
    from dc ab'(1,2) have "a'*?g dvd (b'*?g) *c" by auto
huffman@31706
   874
    hence "?g*a' dvd ?g * (b' * c)" by (simp add: mult_assoc)
huffman@31706
   875
    with z have th_1: "a' dvd b' * c" by auto
huffman@31706
   876
    from int_coprime_dvd_mult[OF ab'(3)] th_1
huffman@31706
   877
    have thc: "a' dvd c" by (subst (asm) mult_commute, blast)
huffman@31706
   878
    from ab' have "a = ?g*a'" by algebra
huffman@31706
   879
    with thb thc have ?thesis by blast }
huffman@31706
   880
  ultimately show ?thesis by blast
chaieb@27669
   881
qed
chaieb@27669
   882
huffman@31706
   883
lemma nat_pow_divides_pow:
huffman@31706
   884
  assumes ab: "(a::nat) ^ n dvd b ^n" and n:"n \<noteq> 0"
huffman@31706
   885
  shows "a dvd b"
huffman@31706
   886
proof-
huffman@31706
   887
  let ?g = "gcd a b"
huffman@31706
   888
  from n obtain m where m: "n = Suc m" by (cases n, simp_all)
huffman@31706
   889
  {assume "?g = 0" with ab n have ?thesis by auto }
huffman@31706
   890
  moreover
huffman@31706
   891
  {assume z: "?g \<noteq> 0"
huffman@31706
   892
    hence zn: "?g ^ n \<noteq> 0" using n by (simp add: neq0_conv)
huffman@31706
   893
    from nat_gcd_coprime_exists[OF z]
huffman@31706
   894
    obtain a' b' where ab': "a = a' * ?g" "b = b' * ?g" "coprime a' b'"
huffman@31706
   895
      by blast
huffman@31706
   896
    from ab have "(a' * ?g) ^ n dvd (b' * ?g)^n"
huffman@31706
   897
      by (simp add: ab'(1,2)[symmetric])
huffman@31706
   898
    hence "?g^n*a'^n dvd ?g^n *b'^n"
huffman@31706
   899
      by (simp only: power_mult_distrib mult_commute)
huffman@31706
   900
    with zn z n have th0:"a'^n dvd b'^n" by auto
huffman@31706
   901
    have "a' dvd a'^n" by (simp add: m)
huffman@31706
   902
    with th0 have "a' dvd b'^n" using dvd_trans[of a' "a'^n" "b'^n"] by simp
huffman@31706
   903
    hence th1: "a' dvd b'^m * b'" by (simp add: m mult_commute)
huffman@31706
   904
    from nat_coprime_dvd_mult[OF nat_coprime_exp [OF ab'(3), of m]] th1
huffman@31706
   905
    have "a' dvd b'" by (subst (asm) mult_commute, blast)
huffman@31706
   906
    hence "a'*?g dvd b'*?g" by simp
huffman@31706
   907
    with ab'(1,2)  have ?thesis by simp }
huffman@31706
   908
  ultimately show ?thesis by blast
huffman@31706
   909
qed
huffman@31706
   910
huffman@31706
   911
lemma int_pow_divides_pow:
huffman@31706
   912
  assumes ab: "(a::int) ^ n dvd b ^n" and n:"n \<noteq> 0"
huffman@31706
   913
  shows "a dvd b"
chaieb@27669
   914
proof-
huffman@31706
   915
  let ?g = "gcd a b"
huffman@31706
   916
  from n obtain m where m: "n = Suc m" by (cases n, simp_all)
huffman@31706
   917
  {assume "?g = 0" with ab n have ?thesis by auto }
huffman@31706
   918
  moreover
huffman@31706
   919
  {assume z: "?g \<noteq> 0"
huffman@31706
   920
    hence zn: "?g ^ n \<noteq> 0" using n by (simp add: neq0_conv)
huffman@31706
   921
    from int_gcd_coprime_exists[OF z]
huffman@31706
   922
    obtain a' b' where ab': "a = a' * ?g" "b = b' * ?g" "coprime a' b'"
huffman@31706
   923
      by blast
huffman@31706
   924
    from ab have "(a' * ?g) ^ n dvd (b' * ?g)^n"
huffman@31706
   925
      by (simp add: ab'(1,2)[symmetric])
huffman@31706
   926
    hence "?g^n*a'^n dvd ?g^n *b'^n"
huffman@31706
   927
      by (simp only: power_mult_distrib mult_commute)
huffman@31706
   928
    with zn z n have th0:"a'^n dvd b'^n" by auto
huffman@31706
   929
    have "a' dvd a'^n" by (simp add: m)
huffman@31706
   930
    with th0 have "a' dvd b'^n"
huffman@31706
   931
      using dvd_trans[of a' "a'^n" "b'^n"] by simp
huffman@31706
   932
    hence th1: "a' dvd b'^m * b'" by (simp add: m mult_commute)
huffman@31706
   933
    from int_coprime_dvd_mult[OF int_coprime_exp [OF ab'(3), of m]] th1
huffman@31706
   934
    have "a' dvd b'" by (subst (asm) mult_commute, blast)
huffman@31706
   935
    hence "a'*?g dvd b'*?g" by simp
huffman@31706
   936
    with ab'(1,2)  have ?thesis by simp }
huffman@31706
   937
  ultimately show ?thesis by blast
huffman@31706
   938
qed
huffman@31706
   939
huffman@31706
   940
lemma nat_pow_divides_eq [simp]: "n ~= 0 \<Longrightarrow> ((a::nat)^n dvd b^n) = (a dvd b)"
huffman@31706
   941
  by (auto intro: nat_pow_divides_pow dvd_power_same)
huffman@31706
   942
huffman@31706
   943
lemma int_pow_divides_eq [simp]: "n ~= 0 \<Longrightarrow> ((a::int)^n dvd b^n) = (a dvd b)"
huffman@31706
   944
  by (auto intro: int_pow_divides_pow dvd_power_same)
huffman@31706
   945
huffman@31706
   946
lemma nat_divides_mult:
huffman@31706
   947
  assumes mr: "(m::nat) dvd r" and nr: "n dvd r" and mn:"coprime m n"
huffman@31706
   948
  shows "m * n dvd r"
huffman@31706
   949
proof-
huffman@31706
   950
  from mr nr obtain m' n' where m': "r = m*m'" and n': "r = n*n'"
huffman@31706
   951
    unfolding dvd_def by blast
huffman@31706
   952
  from mr n' have "m dvd n'*n" by (simp add: mult_commute)
huffman@31706
   953
  hence "m dvd n'" using nat_coprime_dvd_mult_iff[OF mn] by simp
huffman@31706
   954
  then obtain k where k: "n' = m*k" unfolding dvd_def by blast
huffman@31706
   955
  from n' k show ?thesis unfolding dvd_def by auto
huffman@31706
   956
qed
huffman@31706
   957
huffman@31706
   958
lemma int_divides_mult:
huffman@31706
   959
  assumes mr: "(m::int) dvd r" and nr: "n dvd r" and mn:"coprime m n"
huffman@31706
   960
  shows "m * n dvd r"
huffman@31706
   961
proof-
huffman@31706
   962
  from mr nr obtain m' n' where m': "r = m*m'" and n': "r = n*n'"
huffman@31706
   963
    unfolding dvd_def by blast
huffman@31706
   964
  from mr n' have "m dvd n'*n" by (simp add: mult_commute)
huffman@31706
   965
  hence "m dvd n'" using int_coprime_dvd_mult_iff[OF mn] by simp
huffman@31706
   966
  then obtain k where k: "n' = m*k" unfolding dvd_def by blast
huffman@31706
   967
  from n' k show ?thesis unfolding dvd_def by auto
chaieb@27669
   968
qed
chaieb@27669
   969
huffman@31706
   970
lemma nat_coprime_plus_one [simp]: "coprime ((n::nat) + 1) n"
huffman@31706
   971
  apply (subgoal_tac "gcd (n + 1) n dvd (n + 1 - n)")
huffman@31706
   972
  apply force
huffman@31706
   973
  apply (rule nat_dvd_diff)
huffman@31706
   974
  apply auto
huffman@31706
   975
done
huffman@31706
   976
huffman@31706
   977
lemma nat_coprime_Suc [simp]: "coprime (Suc n) n"
huffman@31706
   978
  using nat_coprime_plus_one by (simp add: One_nat_def)
huffman@31706
   979
huffman@31706
   980
lemma int_coprime_plus_one [simp]: "coprime ((n::int) + 1) n"
huffman@31706
   981
  apply (subgoal_tac "gcd (n + 1) n dvd (n + 1 - n)")
huffman@31706
   982
  apply force
huffman@31706
   983
  apply (rule dvd_diff)
huffman@31706
   984
  apply auto
huffman@31706
   985
done
huffman@31706
   986
huffman@31706
   987
lemma nat_coprime_minus_one: "(n::nat) \<noteq> 0 \<Longrightarrow> coprime (n - 1) n"
huffman@31706
   988
  using nat_coprime_plus_one [of "n - 1"]
huffman@31706
   989
    nat_gcd_commute [of "n - 1" n] by auto
huffman@31706
   990
huffman@31706
   991
lemma int_coprime_minus_one: "coprime ((n::int) - 1) n"
huffman@31706
   992
  using int_coprime_plus_one [of "n - 1"]
huffman@31706
   993
    int_gcd_commute [of "n - 1" n] by auto
huffman@31706
   994
huffman@31706
   995
lemma nat_setprod_coprime [rule_format]:
huffman@31706
   996
    "(ALL i: A. coprime (f i) (x::nat)) --> coprime (PROD i:A. f i) x"
huffman@31706
   997
  apply (case_tac "finite A")
huffman@31706
   998
  apply (induct set: finite)
huffman@31706
   999
  apply (auto simp add: nat_gcd_mult_cancel)
huffman@31706
  1000
done
huffman@31706
  1001
huffman@31706
  1002
lemma int_setprod_coprime [rule_format]:
huffman@31706
  1003
    "(ALL i: A. coprime (f i) (x::int)) --> coprime (PROD i:A. f i) x"
huffman@31706
  1004
  apply (case_tac "finite A")
huffman@31706
  1005
  apply (induct set: finite)
huffman@31706
  1006
  apply (auto simp add: int_gcd_mult_cancel)
huffman@31706
  1007
done
huffman@31706
  1008
huffman@31706
  1009
lemma nat_prime_odd: "prime (p::nat) \<Longrightarrow> p > 2 \<Longrightarrow> odd p"
huffman@31706
  1010
  unfolding prime_nat_def
huffman@31706
  1011
  apply (subst even_mult_two_ex)
huffman@31706
  1012
  apply clarify
huffman@31706
  1013
  apply (drule_tac x = 2 in spec)
huffman@31706
  1014
  apply auto
huffman@31706
  1015
done
huffman@31706
  1016
huffman@31706
  1017
lemma int_prime_odd: "prime (p::int) \<Longrightarrow> p > 2 \<Longrightarrow> odd p"
huffman@31706
  1018
  unfolding prime_int_def
huffman@31706
  1019
  apply (frule nat_prime_odd)
huffman@31706
  1020
  apply (auto simp add: even_nat_def)
huffman@31706
  1021
done
huffman@31706
  1022
huffman@31706
  1023
lemma nat_coprime_common_divisor: "coprime (a::nat) b \<Longrightarrow> x dvd a \<Longrightarrow>
huffman@31706
  1024
    x dvd b \<Longrightarrow> x = 1"
huffman@31706
  1025
  apply (subgoal_tac "x dvd gcd a b")
huffman@31706
  1026
  apply simp
huffman@31706
  1027
  apply (erule (1) nat_gcd_greatest)
huffman@31706
  1028
done
huffman@31706
  1029
huffman@31706
  1030
lemma int_coprime_common_divisor: "coprime (a::int) b \<Longrightarrow> x dvd a \<Longrightarrow>
huffman@31706
  1031
    x dvd b \<Longrightarrow> abs x = 1"
huffman@31706
  1032
  apply (subgoal_tac "x dvd gcd a b")
huffman@31706
  1033
  apply simp
huffman@31706
  1034
  apply (erule (1) int_gcd_greatest)
huffman@31706
  1035
done
huffman@31706
  1036
huffman@31706
  1037
lemma nat_coprime_divisors: "(d::int) dvd a \<Longrightarrow> e dvd b \<Longrightarrow> coprime a b \<Longrightarrow>
huffman@31706
  1038
    coprime d e"
huffman@31706
  1039
  apply (auto simp add: dvd_def)
huffman@31706
  1040
  apply (frule int_coprime_lmult)
huffman@31706
  1041
  apply (subst int_gcd_commute)
huffman@31706
  1042
  apply (subst (asm) (2) int_gcd_commute)
huffman@31706
  1043
  apply (erule int_coprime_lmult)
huffman@31706
  1044
done
huffman@31706
  1045
huffman@31706
  1046
lemma nat_invertible_coprime: "(x::nat) * y mod m = 1 \<Longrightarrow> coprime x m"
huffman@31706
  1047
apply (metis nat_coprime_lmult nat_gcd_1 nat_gcd_commute nat_gcd_red)
huffman@31706
  1048
done
huffman@31706
  1049
huffman@31706
  1050
lemma int_invertible_coprime: "(x::int) * y mod m = 1 \<Longrightarrow> coprime x m"
huffman@31706
  1051
apply (metis int_coprime_lmult int_gcd_1 int_gcd_commute int_gcd_red)
huffman@31706
  1052
done
huffman@31706
  1053
huffman@31706
  1054
huffman@31706
  1055
subsection {* Bezout's theorem *}
huffman@31706
  1056
huffman@31706
  1057
(* Function bezw returns a pair of witnesses to Bezout's theorem --
huffman@31706
  1058
   see the theorems that follow the definition. *)
huffman@31706
  1059
fun
huffman@31706
  1060
  bezw  :: "nat \<Rightarrow> nat \<Rightarrow> int * int"
huffman@31706
  1061
where
huffman@31706
  1062
  "bezw x y =
huffman@31706
  1063
  (if y = 0 then (1, 0) else
huffman@31706
  1064
      (snd (bezw y (x mod y)),
huffman@31706
  1065
       fst (bezw y (x mod y)) - snd (bezw y (x mod y)) * int(x div y)))"
huffman@31706
  1066
huffman@31706
  1067
lemma bezw_0 [simp]: "bezw x 0 = (1, 0)" by simp
huffman@31706
  1068
huffman@31706
  1069
lemma bezw_non_0: "y > 0 \<Longrightarrow> bezw x y = (snd (bezw y (x mod y)),
huffman@31706
  1070
       fst (bezw y (x mod y)) - snd (bezw y (x mod y)) * int(x div y))"
huffman@31706
  1071
  by simp
huffman@31706
  1072
huffman@31706
  1073
declare bezw.simps [simp del]
huffman@31706
  1074
huffman@31706
  1075
lemma bezw_aux [rule_format]:
huffman@31706
  1076
    "fst (bezw x y) * int x + snd (bezw x y) * int y = int (gcd x y)"
huffman@31706
  1077
proof (induct x y rule: nat_gcd_induct)
huffman@31706
  1078
  fix m :: nat
huffman@31706
  1079
  show "fst (bezw m 0) * int m + snd (bezw m 0) * int 0 = int (gcd m 0)"
huffman@31706
  1080
    by auto
huffman@31706
  1081
  next fix m :: nat and n
huffman@31706
  1082
    assume ngt0: "n > 0" and
huffman@31706
  1083
      ih: "fst (bezw n (m mod n)) * int n +
huffman@31706
  1084
        snd (bezw n (m mod n)) * int (m mod n) =
huffman@31706
  1085
        int (gcd n (m mod n))"
huffman@31706
  1086
    thus "fst (bezw m n) * int m + snd (bezw m n) * int n = int (gcd m n)"
huffman@31706
  1087
      apply (simp add: bezw_non_0 nat_gcd_non_0)
huffman@31706
  1088
      apply (erule subst)
huffman@31706
  1089
      apply (simp add: ring_simps)
huffman@31706
  1090
      apply (subst mod_div_equality [of m n, symmetric])
huffman@31706
  1091
      (* applying simp here undoes the last substitution!
huffman@31706
  1092
         what is procedure cancel_div_mod? *)
huffman@31706
  1093
      apply (simp only: ring_simps zadd_int [symmetric]
huffman@31706
  1094
        zmult_int [symmetric])
huffman@31706
  1095
      done
huffman@31706
  1096
qed
huffman@31706
  1097
huffman@31706
  1098
lemma int_bezout:
huffman@31706
  1099
  fixes x y
huffman@31706
  1100
  shows "EX u v. u * (x::int) + v * y = gcd x y"
huffman@31706
  1101
proof -
huffman@31706
  1102
  have bezout_aux: "!!x y. x \<ge> (0::int) \<Longrightarrow> y \<ge> 0 \<Longrightarrow>
huffman@31706
  1103
      EX u v. u * x + v * y = gcd x y"
huffman@31706
  1104
    apply (rule_tac x = "fst (bezw (nat x) (nat y))" in exI)
huffman@31706
  1105
    apply (rule_tac x = "snd (bezw (nat x) (nat y))" in exI)
huffman@31706
  1106
    apply (unfold gcd_int_def)
huffman@31706
  1107
    apply simp
huffman@31706
  1108
    apply (subst bezw_aux [symmetric])
huffman@31706
  1109
    apply auto
huffman@31706
  1110
    done
huffman@31706
  1111
  have "(x \<ge> 0 \<and> y \<ge> 0) | (x \<ge> 0 \<and> y \<le> 0) | (x \<le> 0 \<and> y \<ge> 0) |
huffman@31706
  1112
      (x \<le> 0 \<and> y \<le> 0)"
huffman@31706
  1113
    by auto
huffman@31706
  1114
  moreover have "x \<ge> 0 \<Longrightarrow> y \<ge> 0 \<Longrightarrow> ?thesis"
huffman@31706
  1115
    by (erule (1) bezout_aux)
huffman@31706
  1116
  moreover have "x >= 0 \<Longrightarrow> y <= 0 \<Longrightarrow> ?thesis"
huffman@31706
  1117
    apply (insert bezout_aux [of x "-y"])
huffman@31706
  1118
    apply auto
huffman@31706
  1119
    apply (rule_tac x = u in exI)
huffman@31706
  1120
    apply (rule_tac x = "-v" in exI)
huffman@31706
  1121
    apply (subst int_gcd_neg2 [symmetric])
huffman@31706
  1122
    apply auto
huffman@31706
  1123
    done
huffman@31706
  1124
  moreover have "x <= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> ?thesis"
huffman@31706
  1125
    apply (insert bezout_aux [of "-x" y])
huffman@31706
  1126
    apply auto
huffman@31706
  1127
    apply (rule_tac x = "-u" in exI)
huffman@31706
  1128
    apply (rule_tac x = v in exI)
huffman@31706
  1129
    apply (subst int_gcd_neg1 [symmetric])
huffman@31706
  1130
    apply auto
huffman@31706
  1131
    done
huffman@31706
  1132
  moreover have "x <= 0 \<Longrightarrow> y <= 0 \<Longrightarrow> ?thesis"
huffman@31706
  1133
    apply (insert bezout_aux [of "-x" "-y"])
huffman@31706
  1134
    apply auto
huffman@31706
  1135
    apply (rule_tac x = "-u" in exI)
huffman@31706
  1136
    apply (rule_tac x = "-v" in exI)
huffman@31706
  1137
    apply (subst int_gcd_neg1 [symmetric])
huffman@31706
  1138
    apply (subst int_gcd_neg2 [symmetric])
huffman@31706
  1139
    apply auto
huffman@31706
  1140
    done
huffman@31706
  1141
  ultimately show ?thesis by blast
huffman@31706
  1142
qed
huffman@31706
  1143
huffman@31706
  1144
text {* versions of Bezout for nat, by Amine Chaieb *}
huffman@31706
  1145
huffman@31706
  1146
lemma ind_euclid:
huffman@31706
  1147
  assumes c: " \<forall>a b. P (a::nat) b \<longleftrightarrow> P b a" and z: "\<forall>a. P a 0"
huffman@31706
  1148
  and add: "\<forall>a b. P a b \<longrightarrow> P a (a + b)"
chaieb@27669
  1149
  shows "P a b"
chaieb@27669
  1150
proof(induct n\<equiv>"a+b" arbitrary: a b rule: nat_less_induct)
chaieb@27669
  1151
  fix n a b
chaieb@27669
  1152
  assume H: "\<forall>m < n. \<forall>a b. m = a + b \<longrightarrow> P a b" "n = a + b"
chaieb@27669
  1153
  have "a = b \<or> a < b \<or> b < a" by arith
chaieb@27669
  1154
  moreover {assume eq: "a= b"
huffman@31706
  1155
    from add[rule_format, OF z[rule_format, of a]] have "P a b" using eq
huffman@31706
  1156
    by simp}
chaieb@27669
  1157
  moreover
chaieb@27669
  1158
  {assume lt: "a < b"
chaieb@27669
  1159
    hence "a + b - a < n \<or> a = 0"  using H(2) by arith
chaieb@27669
  1160
    moreover
chaieb@27669
  1161
    {assume "a =0" with z c have "P a b" by blast }
chaieb@27669
  1162
    moreover
chaieb@27669
  1163
    {assume ab: "a + b - a < n"
chaieb@27669
  1164
      have th0: "a + b - a = a + (b - a)" using lt by arith
chaieb@27669
  1165
      from add[rule_format, OF H(1)[rule_format, OF ab th0]]
chaieb@27669
  1166
      have "P a b" by (simp add: th0[symmetric])}
chaieb@27669
  1167
    ultimately have "P a b" by blast}
chaieb@27669
  1168
  moreover
chaieb@27669
  1169
  {assume lt: "a > b"
chaieb@27669
  1170
    hence "b + a - b < n \<or> b = 0"  using H(2) by arith
chaieb@27669
  1171
    moreover
chaieb@27669
  1172
    {assume "b =0" with z c have "P a b" by blast }
chaieb@27669
  1173
    moreover
chaieb@27669
  1174
    {assume ab: "b + a - b < n"
chaieb@27669
  1175
      have th0: "b + a - b = b + (a - b)" using lt by arith
chaieb@27669
  1176
      from add[rule_format, OF H(1)[rule_format, OF ab th0]]
chaieb@27669
  1177
      have "P b a" by (simp add: th0[symmetric])
chaieb@27669
  1178
      hence "P a b" using c by blast }
chaieb@27669
  1179
    ultimately have "P a b" by blast}
chaieb@27669
  1180
ultimately  show "P a b" by blast
chaieb@27669
  1181
qed
chaieb@27669
  1182
huffman@31706
  1183
lemma nat_bezout_lemma:
huffman@31706
  1184
  assumes ex: "\<exists>(d::nat) x y. d dvd a \<and> d dvd b \<and>
huffman@31706
  1185
    (a * x = b * y + d \<or> b * x = a * y + d)"
huffman@31706
  1186
  shows "\<exists>d x y. d dvd a \<and> d dvd a + b \<and>
huffman@31706
  1187
    (a * x = (a + b) * y + d \<or> (a + b) * x = a * y + d)"
huffman@31706
  1188
  using ex
huffman@31706
  1189
  apply clarsimp
huffman@31706
  1190
  apply (rule_tac x="d" in exI, simp add: dvd_add)
huffman@31706
  1191
  apply (case_tac "a * x = b * y + d" , simp_all)
huffman@31706
  1192
  apply (rule_tac x="x + y" in exI)
huffman@31706
  1193
  apply (rule_tac x="y" in exI)
huffman@31706
  1194
  apply algebra
huffman@31706
  1195
  apply (rule_tac x="x" in exI)
huffman@31706
  1196
  apply (rule_tac x="x + y" in exI)
huffman@31706
  1197
  apply algebra
chaieb@27669
  1198
done
chaieb@27669
  1199
huffman@31706
  1200
lemma nat_bezout_add: "\<exists>(d::nat) x y. d dvd a \<and> d dvd b \<and>
huffman@31706
  1201
    (a * x = b * y + d \<or> b * x = a * y + d)"
huffman@31706
  1202
  apply(induct a b rule: ind_euclid)
huffman@31706
  1203
  apply blast
huffman@31706
  1204
  apply clarify
huffman@31706
  1205
  apply (rule_tac x="a" in exI, simp add: dvd_add)
huffman@31706
  1206
  apply clarsimp
huffman@31706
  1207
  apply (rule_tac x="d" in exI)
huffman@31706
  1208
  apply (case_tac "a * x = b * y + d", simp_all add: dvd_add)
huffman@31706
  1209
  apply (rule_tac x="x+y" in exI)
huffman@31706
  1210
  apply (rule_tac x="y" in exI)
huffman@31706
  1211
  apply algebra
huffman@31706
  1212
  apply (rule_tac x="x" in exI)
huffman@31706
  1213
  apply (rule_tac x="x+y" in exI)
huffman@31706
  1214
  apply algebra
chaieb@27669
  1215
done
chaieb@27669
  1216
huffman@31706
  1217
lemma nat_bezout1: "\<exists>(d::nat) x y. d dvd a \<and> d dvd b \<and>
huffman@31706
  1218
    (a * x - b * y = d \<or> b * x - a * y = d)"
huffman@31706
  1219
  using nat_bezout_add[of a b]
huffman@31706
  1220
  apply clarsimp
huffman@31706
  1221
  apply (rule_tac x="d" in exI, simp)
huffman@31706
  1222
  apply (rule_tac x="x" in exI)
huffman@31706
  1223
  apply (rule_tac x="y" in exI)
huffman@31706
  1224
  apply auto
chaieb@27669
  1225
done
chaieb@27669
  1226
huffman@31706
  1227
lemma nat_bezout_add_strong: assumes nz: "a \<noteq> (0::nat)"
chaieb@27669
  1228
  shows "\<exists>d x y. d dvd a \<and> d dvd b \<and> a * x = b * y + d"
chaieb@27669
  1229
proof-
huffman@31706
  1230
 from nz have ap: "a > 0" by simp
huffman@31706
  1231
 from nat_bezout_add[of a b]
huffman@31706
  1232
 have "(\<exists>d x y. d dvd a \<and> d dvd b \<and> a * x = b * y + d) \<or>
huffman@31706
  1233
   (\<exists>d x y. d dvd a \<and> d dvd b \<and> b * x = a * y + d)" by blast
chaieb@27669
  1234
 moreover
huffman@31706
  1235
    {fix d x y assume H: "d dvd a" "d dvd b" "a * x = b * y + d"
huffman@31706
  1236
     from H have ?thesis by blast }
chaieb@27669
  1237
 moreover
chaieb@27669
  1238
 {fix d x y assume H: "d dvd a" "d dvd b" "b * x = a * y + d"
chaieb@27669
  1239
   {assume b0: "b = 0" with H  have ?thesis by simp}
huffman@31706
  1240
   moreover
chaieb@27669
  1241
   {assume b: "b \<noteq> 0" hence bp: "b > 0" by simp
huffman@31706
  1242
     from b dvd_imp_le [OF H(2)] have "d < b \<or> d = b"
huffman@31706
  1243
       by auto
chaieb@27669
  1244
     moreover
chaieb@27669
  1245
     {assume db: "d=b"
chaieb@27669
  1246
       from prems have ?thesis apply simp
chaieb@27669
  1247
	 apply (rule exI[where x = b], simp)
chaieb@27669
  1248
	 apply (rule exI[where x = b])
chaieb@27669
  1249
	by (rule exI[where x = "a - 1"], simp add: diff_mult_distrib2)}
chaieb@27669
  1250
    moreover
huffman@31706
  1251
    {assume db: "d < b"
chaieb@27669
  1252
	{assume "x=0" hence ?thesis  using prems by simp }
chaieb@27669
  1253
	moreover
chaieb@27669
  1254
	{assume x0: "x \<noteq> 0" hence xp: "x > 0" by simp
chaieb@27669
  1255
	  from db have "d \<le> b - 1" by simp
chaieb@27669
  1256
	  hence "d*b \<le> b*(b - 1)" by simp
chaieb@27669
  1257
	  with xp mult_mono[of "1" "x" "d*b" "b*(b - 1)"]
chaieb@27669
  1258
	  have dble: "d*b \<le> x*b*(b - 1)" using bp by simp
huffman@31706
  1259
	  from H (3) have "d + (b - 1) * (b*x) = d + (b - 1) * (a*y + d)"
huffman@31706
  1260
            by simp
huffman@31706
  1261
	  hence "d + (b - 1) * a * y + (b - 1) * d = d + (b - 1) * b * x"
huffman@31706
  1262
	    by (simp only: mult_assoc right_distrib)
huffman@31706
  1263
	  hence "a * ((b - 1) * y) + d * (b - 1 + 1) = d + x*b*(b - 1)"
huffman@31706
  1264
            by algebra
chaieb@27669
  1265
	  hence "a * ((b - 1) * y) = d + x*b*(b - 1) - d*b" using bp by simp
huffman@31706
  1266
	  hence "a * ((b - 1) * y) = d + (x*b*(b - 1) - d*b)"
chaieb@27669
  1267
	    by (simp only: diff_add_assoc[OF dble, of d, symmetric])
chaieb@27669
  1268
	  hence "a * ((b - 1) * y) = b*(x*(b - 1) - d) + d"
chaieb@27669
  1269
	    by (simp only: diff_mult_distrib2 add_commute mult_ac)
chaieb@27669
  1270
	  hence ?thesis using H(1,2)
chaieb@27669
  1271
	    apply -
chaieb@27669
  1272
	    apply (rule exI[where x=d], simp)
chaieb@27669
  1273
	    apply (rule exI[where x="(b - 1) * y"])
chaieb@27669
  1274
	    by (rule exI[where x="x*(b - 1) - d"], simp)}
chaieb@27669
  1275
	ultimately have ?thesis by blast}
chaieb@27669
  1276
    ultimately have ?thesis by blast}
chaieb@27669
  1277
  ultimately have ?thesis by blast}
chaieb@27669
  1278
 ultimately show ?thesis by blast
chaieb@27669
  1279
qed
chaieb@27669
  1280
huffman@31706
  1281
lemma nat_bezout: assumes a: "(a::nat) \<noteq> 0"
chaieb@27669
  1282
  shows "\<exists>x y. a * x = b * y + gcd a b"
chaieb@27669
  1283
proof-
chaieb@27669
  1284
  let ?g = "gcd a b"
huffman@31706
  1285
  from nat_bezout_add_strong[OF a, of b]
chaieb@27669
  1286
  obtain d x y where d: "d dvd a" "d dvd b" "a * x = b * y + d" by blast
chaieb@27669
  1287
  from d(1,2) have "d dvd ?g" by simp
chaieb@27669
  1288
  then obtain k where k: "?g = d*k" unfolding dvd_def by blast
huffman@31706
  1289
  from d(3) have "a * x * k = (b * y + d) *k " by auto
chaieb@27669
  1290
  hence "a * (x * k) = b * (y*k) + ?g" by (algebra add: k)
chaieb@27669
  1291
  thus ?thesis by blast
chaieb@27669
  1292
qed
chaieb@27669
  1293
huffman@31706
  1294
huffman@31706
  1295
subsection {* LCM *}
huffman@31706
  1296
huffman@31706
  1297
lemma int_lcm_altdef: "lcm (a::int) b = (abs a) * (abs b) div gcd a b"
huffman@31706
  1298
  by (simp add: lcm_int_def lcm_nat_def zdiv_int
huffman@31706
  1299
    zmult_int [symmetric] gcd_int_def)
huffman@31706
  1300
huffman@31706
  1301
lemma nat_prod_gcd_lcm: "(m::nat) * n = gcd m n * lcm m n"
huffman@31706
  1302
  unfolding lcm_nat_def
huffman@31706
  1303
  by (simp add: dvd_mult_div_cancel [OF nat_gcd_dvd_prod])
huffman@31706
  1304
huffman@31706
  1305
lemma int_prod_gcd_lcm: "abs(m::int) * abs n = gcd m n * lcm m n"
huffman@31706
  1306
  unfolding lcm_int_def gcd_int_def
huffman@31706
  1307
  apply (subst int_mult [symmetric])
huffman@31706
  1308
  apply (subst nat_prod_gcd_lcm [symmetric])
huffman@31706
  1309
  apply (subst nat_abs_mult_distrib [symmetric])
huffman@31706
  1310
  apply (simp, simp add: abs_mult)
huffman@31706
  1311
done
huffman@31706
  1312
huffman@31706
  1313
lemma nat_lcm_0 [simp]: "lcm (m::nat) 0 = 0"
huffman@31706
  1314
  unfolding lcm_nat_def by simp
huffman@31706
  1315
huffman@31706
  1316
lemma int_lcm_0 [simp]: "lcm (m::int) 0 = 0"
huffman@31706
  1317
  unfolding lcm_int_def by simp
huffman@31706
  1318
huffman@31706
  1319
lemma nat_lcm_1 [simp]: "lcm (m::nat) 1 = m"
huffman@31706
  1320
  unfolding lcm_nat_def by simp
huffman@31706
  1321
huffman@31706
  1322
lemma nat_lcm_Suc_0 [simp]: "lcm (m::nat) (Suc 0) = m"
huffman@31706
  1323
  unfolding lcm_nat_def by (simp add: One_nat_def)
huffman@31706
  1324
huffman@31706
  1325
lemma int_lcm_1 [simp]: "lcm (m::int) 1 = abs m"
huffman@31706
  1326
  unfolding lcm_int_def by simp
huffman@31706
  1327
huffman@31706
  1328
lemma nat_lcm_0_left [simp]: "lcm (0::nat) n = 0"
huffman@31706
  1329
  unfolding lcm_nat_def by simp
chaieb@27669
  1330
huffman@31706
  1331
lemma int_lcm_0_left [simp]: "lcm (0::int) n = 0"
huffman@31706
  1332
  unfolding lcm_int_def by simp
huffman@31706
  1333
huffman@31706
  1334
lemma nat_lcm_1_left [simp]: "lcm (1::nat) m = m"
huffman@31706
  1335
  unfolding lcm_nat_def by simp
huffman@31706
  1336
huffman@31706
  1337
lemma nat_lcm_Suc_0_left [simp]: "lcm (Suc 0) m = m"
huffman@31706
  1338
  unfolding lcm_nat_def by (simp add: One_nat_def)
huffman@31706
  1339
huffman@31706
  1340
lemma int_lcm_1_left [simp]: "lcm (1::int) m = abs m"
huffman@31706
  1341
  unfolding lcm_int_def by simp
huffman@31706
  1342
huffman@31706
  1343
lemma nat_lcm_commute: "lcm (m::nat) n = lcm n m"
huffman@31706
  1344
  unfolding lcm_nat_def by (simp add: nat_gcd_commute ring_simps)
huffman@31706
  1345
huffman@31706
  1346
lemma int_lcm_commute: "lcm (m::int) n = lcm n m"
huffman@31706
  1347
  unfolding lcm_int_def by (subst nat_lcm_commute, rule refl)
huffman@31706
  1348
huffman@31706
  1349
huffman@31706
  1350
lemma nat_lcm_pos:
huffman@31706
  1351
  assumes mpos: "(m::nat) > 0"
huffman@31706
  1352
  and npos: "n>0"
huffman@31706
  1353
  shows "lcm m n > 0"
huffman@31706
  1354
proof(rule ccontr, simp add: lcm_nat_def nat_gcd_zero)
huffman@31706
  1355
  assume h:"m*n div gcd m n = 0"
huffman@31706
  1356
  from mpos npos have "gcd m n \<noteq> 0" using nat_gcd_zero by simp
huffman@31706
  1357
  hence gcdp: "gcd m n > 0" by simp
huffman@31706
  1358
  with h
huffman@31706
  1359
  have "m*n < gcd m n"
huffman@31706
  1360
    by (cases "m * n < gcd m n")
huffman@31706
  1361
       (auto simp add: div_if[OF gcdp, where m="m*n"])
chaieb@27669
  1362
  moreover
huffman@31706
  1363
  have "gcd m n dvd m" by simp
huffman@31706
  1364
  with mpos dvd_imp_le have t1:"gcd m n \<le> m" by simp
huffman@31706
  1365
  with npos have t1:"gcd m n*n \<le> m*n" by simp
huffman@31706
  1366
  have "gcd m n \<le> gcd m n*n" using npos by simp
huffman@31706
  1367
  with t1 have "gcd m n \<le> m*n" by arith
huffman@31706
  1368
  ultimately show "False" by simp
chaieb@27669
  1369
qed
chaieb@27669
  1370
huffman@31706
  1371
lemma int_lcm_pos:
huffman@31706
  1372
  assumes mneq0: "(m::int) ~= 0"
huffman@31706
  1373
  and npos: "n ~= 0"
huffman@31706
  1374
  shows "lcm m n > 0"
chaieb@27669
  1375
huffman@31706
  1376
  apply (subst int_lcm_abs)
huffman@31706
  1377
  apply (rule nat_lcm_pos [transferred])
huffman@31706
  1378
  using prems apply auto
huffman@31706
  1379
done
haftmann@23687
  1380
huffman@31706
  1381
lemma nat_dvd_pos:
haftmann@23687
  1382
  fixes n m :: nat
haftmann@23687
  1383
  assumes "n > 0" and "m dvd n"
haftmann@23687
  1384
  shows "m > 0"
haftmann@23687
  1385
using assms by (cases m) auto
haftmann@23687
  1386
huffman@31706
  1387
lemma nat_lcm_least:
huffman@31706
  1388
  assumes "(m::nat) dvd k" and "n dvd k"
haftmann@27556
  1389
  shows "lcm m n dvd k"
haftmann@23687
  1390
proof (cases k)
haftmann@23687
  1391
  case 0 then show ?thesis by auto
haftmann@23687
  1392
next
haftmann@23687
  1393
  case (Suc _) then have pos_k: "k > 0" by auto
huffman@31706
  1394
  from assms nat_dvd_pos [OF this] have pos_mn: "m > 0" "n > 0" by auto
huffman@31706
  1395
  with nat_gcd_zero [of m n] have pos_gcd: "gcd m n > 0" by simp
haftmann@23687
  1396
  from assms obtain p where k_m: "k = m * p" using dvd_def by blast
haftmann@23687
  1397
  from assms obtain q where k_n: "k = n * q" using dvd_def by blast
haftmann@23687
  1398
  from pos_k k_m have pos_p: "p > 0" by auto
haftmann@23687
  1399
  from pos_k k_n have pos_q: "q > 0" by auto
haftmann@27556
  1400
  have "k * k * gcd q p = k * gcd (k * q) (k * p)"
huffman@31706
  1401
    by (simp add: mult_ac nat_gcd_mult_distrib)
haftmann@27556
  1402
  also have "\<dots> = k * gcd (m * p * q) (n * q * p)"
haftmann@23687
  1403
    by (simp add: k_m [symmetric] k_n [symmetric])
haftmann@27556
  1404
  also have "\<dots> = k * p * q * gcd m n"
huffman@31706
  1405
    by (simp add: mult_ac nat_gcd_mult_distrib)
haftmann@27556
  1406
  finally have "(m * p) * (n * q) * gcd q p = k * p * q * gcd m n"
haftmann@23687
  1407
    by (simp only: k_m [symmetric] k_n [symmetric])
haftmann@27556
  1408
  then have "p * q * m * n * gcd q p = p * q * k * gcd m n"
haftmann@23687
  1409
    by (simp add: mult_ac)
haftmann@27556
  1410
  with pos_p pos_q have "m * n * gcd q p = k * gcd m n"
haftmann@23687
  1411
    by simp
huffman@31706
  1412
  with nat_prod_gcd_lcm [of m n]
haftmann@27556
  1413
  have "lcm m n * gcd q p * gcd m n = k * gcd m n"
haftmann@23687
  1414
    by (simp add: mult_ac)
huffman@31706
  1415
  with pos_gcd have "lcm m n * gcd q p = k" by auto
haftmann@23687
  1416
  then show ?thesis using dvd_def by auto
haftmann@23687
  1417
qed
haftmann@23687
  1418
huffman@31706
  1419
lemma int_lcm_least:
huffman@31706
  1420
  assumes "(m::int) dvd k" and "n dvd k"
huffman@31706
  1421
  shows "lcm m n dvd k"
huffman@31706
  1422
huffman@31706
  1423
  apply (subst int_lcm_abs)
huffman@31706
  1424
  apply (rule dvd_trans)
huffman@31706
  1425
  apply (rule nat_lcm_least [transferred, of _ "abs k" _])
huffman@31706
  1426
  using prems apply auto
huffman@31706
  1427
done
huffman@31706
  1428
nipkow@31730
  1429
lemma nat_lcm_dvd1: "(m::nat) dvd lcm m n"
haftmann@23687
  1430
proof (cases m)
haftmann@23687
  1431
  case 0 then show ?thesis by simp
haftmann@23687
  1432
next
haftmann@23687
  1433
  case (Suc _)
haftmann@23687
  1434
  then have mpos: "m > 0" by simp
haftmann@23687
  1435
  show ?thesis
haftmann@23687
  1436
  proof (cases n)
haftmann@23687
  1437
    case 0 then show ?thesis by simp
haftmann@23687
  1438
  next
haftmann@23687
  1439
    case (Suc _)
haftmann@23687
  1440
    then have npos: "n > 0" by simp
haftmann@27556
  1441
    have "gcd m n dvd n" by simp
haftmann@27556
  1442
    then obtain k where "n = gcd m n * k" using dvd_def by auto
huffman@31706
  1443
    then have "m * n div gcd m n = m * (gcd m n * k) div gcd m n"
huffman@31706
  1444
      by (simp add: mult_ac)
huffman@31706
  1445
    also have "\<dots> = m * k" using mpos npos nat_gcd_zero by simp
huffman@31706
  1446
    finally show ?thesis by (simp add: lcm_nat_def)
haftmann@23687
  1447
  qed
haftmann@23687
  1448
qed
haftmann@23687
  1449
nipkow@31730
  1450
lemma int_lcm_dvd1: "(m::int) dvd lcm m n"
huffman@31706
  1451
  apply (subst int_lcm_abs)
huffman@31706
  1452
  apply (rule dvd_trans)
huffman@31706
  1453
  prefer 2
huffman@31706
  1454
  apply (rule nat_lcm_dvd1 [transferred])
huffman@31706
  1455
  apply auto
huffman@31706
  1456
done
huffman@31706
  1457
nipkow@31730
  1458
lemma nat_lcm_dvd2: "(n::nat) dvd lcm m n"
huffman@31706
  1459
  by (subst nat_lcm_commute, rule nat_lcm_dvd1)
huffman@31706
  1460
nipkow@31730
  1461
lemma int_lcm_dvd2: "(n::int) dvd lcm m n"
huffman@31706
  1462
  by (subst int_lcm_commute, rule int_lcm_dvd1)
huffman@31706
  1463
nipkow@31730
  1464
lemma dvd_lcm_I1_nat[simp]: "(k::nat) dvd m \<Longrightarrow> k dvd lcm m n"
nipkow@31729
  1465
by(metis nat_lcm_dvd1 dvd_trans)
nipkow@31729
  1466
nipkow@31730
  1467
lemma dvd_lcm_I2_nat[simp]: "(k::nat) dvd n \<Longrightarrow> k dvd lcm m n"
nipkow@31729
  1468
by(metis nat_lcm_dvd2 dvd_trans)
nipkow@31729
  1469
nipkow@31730
  1470
lemma dvd_lcm_I1_int[simp]: "(i::int) dvd m \<Longrightarrow> i dvd lcm m n"
nipkow@31729
  1471
by(metis int_lcm_dvd1 dvd_trans)
nipkow@31729
  1472
nipkow@31730
  1473
lemma dvd_lcm_I2_int[simp]: "(i::int) dvd n \<Longrightarrow> i dvd lcm m n"
nipkow@31729
  1474
by(metis int_lcm_dvd2 dvd_trans)
nipkow@31729
  1475
huffman@31706
  1476
lemma nat_lcm_unique: "(a::nat) dvd d \<and> b dvd d \<and>
huffman@31706
  1477
    (\<forall>e. a dvd e \<and> b dvd e \<longrightarrow> d dvd e) \<longleftrightarrow> d = lcm a b"
nipkow@31730
  1478
  by (auto intro: dvd_anti_sym nat_lcm_least nat_lcm_dvd1 nat_lcm_dvd2)
chaieb@27568
  1479
huffman@31706
  1480
lemma int_lcm_unique: "d >= 0 \<and> (a::int) dvd d \<and> b dvd d \<and>
huffman@31706
  1481
    (\<forall>e. a dvd e \<and> b dvd e \<longrightarrow> d dvd e) \<longleftrightarrow> d = lcm a b"
huffman@31706
  1482
  by (auto intro: dvd_anti_sym [transferred] int_lcm_least)
huffman@31706
  1483
huffman@31706
  1484
lemma nat_lcm_dvd_eq [simp]: "(x::nat) dvd y \<Longrightarrow> lcm x y = y"
huffman@31706
  1485
  apply (rule sym)
huffman@31706
  1486
  apply (subst nat_lcm_unique [symmetric])
huffman@31706
  1487
  apply auto
huffman@31706
  1488
done
huffman@31706
  1489
huffman@31706
  1490
lemma int_lcm_dvd_eq [simp]: "0 <= y \<Longrightarrow> (x::int) dvd y \<Longrightarrow> lcm x y = y"
huffman@31706
  1491
  apply (rule sym)
huffman@31706
  1492
  apply (subst int_lcm_unique [symmetric])
huffman@31706
  1493
  apply auto
huffman@31706
  1494
done
huffman@31706
  1495
huffman@31706
  1496
lemma nat_lcm_dvd_eq' [simp]: "(x::nat) dvd y \<Longrightarrow> lcm y x = y"
huffman@31706
  1497
  by (subst nat_lcm_commute, erule nat_lcm_dvd_eq)
huffman@31706
  1498
huffman@31706
  1499
lemma int_lcm_dvd_eq' [simp]: "y >= 0 \<Longrightarrow> (x::int) dvd y \<Longrightarrow> lcm y x = y"
huffman@31706
  1500
  by (subst int_lcm_commute, erule (1) int_lcm_dvd_eq)
huffman@31706
  1501
chaieb@27568
  1502
nipkow@31766
  1503
lemma lcm_assoc_nat: "lcm (lcm n m) (p::nat) = lcm n (lcm m p)"
nipkow@31766
  1504
apply(rule nat_lcm_unique[THEN iffD1])
nipkow@31766
  1505
apply (metis dvd.order_trans nat_lcm_unique)
nipkow@31766
  1506
done
nipkow@31766
  1507
nipkow@31766
  1508
lemma lcm_assoc_int: "lcm (lcm n m) (p::int) = lcm n (lcm m p)"
nipkow@31766
  1509
apply(rule int_lcm_unique[THEN iffD1])
nipkow@31766
  1510
apply (metis dvd_trans int_lcm_unique)
nipkow@31766
  1511
done
nipkow@31766
  1512
nipkow@31766
  1513
lemmas lcm_left_commute_nat =
nipkow@31766
  1514
  mk_left_commute[of lcm, OF lcm_assoc_nat nat_lcm_commute]
nipkow@31766
  1515
nipkow@31766
  1516
lemmas lcm_left_commute_int =
nipkow@31766
  1517
  mk_left_commute[of lcm, OF lcm_assoc_int int_lcm_commute]
nipkow@31766
  1518
nipkow@31766
  1519
lemmas lcm_ac_nat = lcm_assoc_nat nat_lcm_commute lcm_left_commute_nat
nipkow@31766
  1520
lemmas lcm_ac_int = lcm_assoc_int int_lcm_commute lcm_left_commute_int
nipkow@31766
  1521
haftmann@23687
  1522
huffman@31706
  1523
subsection {* Primes *}
wenzelm@22367
  1524
huffman@31706
  1525
(* Is there a better way to handle these, rather than making them
huffman@31706
  1526
   elim rules? *)
chaieb@22027
  1527
huffman@31706
  1528
lemma nat_prime_ge_0 [elim]: "prime (p::nat) \<Longrightarrow> p >= 0"
huffman@31706
  1529
  by (unfold prime_nat_def, auto)
chaieb@22027
  1530
huffman@31706
  1531
lemma nat_prime_gt_0 [elim]: "prime (p::nat) \<Longrightarrow> p > 0"
huffman@31706
  1532
  by (unfold prime_nat_def, auto)
wenzelm@22367
  1533
huffman@31706
  1534
lemma nat_prime_ge_1 [elim]: "prime (p::nat) \<Longrightarrow> p >= 1"
huffman@31706
  1535
  by (unfold prime_nat_def, auto)
chaieb@22027
  1536
huffman@31706
  1537
lemma nat_prime_gt_1 [elim]: "prime (p::nat) \<Longrightarrow> p > 1"
huffman@31706
  1538
  by (unfold prime_nat_def, auto)
wenzelm@22367
  1539
huffman@31706
  1540
lemma nat_prime_ge_Suc_0 [elim]: "prime (p::nat) \<Longrightarrow> p >= Suc 0"
huffman@31706
  1541
  by (unfold prime_nat_def, auto)
wenzelm@22367
  1542
huffman@31706
  1543
lemma nat_prime_gt_Suc_0 [elim]: "prime (p::nat) \<Longrightarrow> p > Suc 0"
huffman@31706
  1544
  by (unfold prime_nat_def, auto)
huffman@31706
  1545
huffman@31706
  1546
lemma nat_prime_ge_2 [elim]: "prime (p::nat) \<Longrightarrow> p >= 2"
huffman@31706
  1547
  by (unfold prime_nat_def, auto)
huffman@31706
  1548
huffman@31706
  1549
lemma int_prime_ge_0 [elim]: "prime (p::int) \<Longrightarrow> p >= 0"
huffman@31706
  1550
  by (unfold prime_int_def prime_nat_def, auto)
wenzelm@22367
  1551
huffman@31706
  1552
lemma int_prime_gt_0 [elim]: "prime (p::int) \<Longrightarrow> p > 0"
huffman@31706
  1553
  by (unfold prime_int_def prime_nat_def, auto)
huffman@31706
  1554
huffman@31706
  1555
lemma int_prime_ge_1 [elim]: "prime (p::int) \<Longrightarrow> p >= 1"
huffman@31706
  1556
  by (unfold prime_int_def prime_nat_def, auto)
chaieb@22027
  1557
huffman@31706
  1558
lemma int_prime_gt_1 [elim]: "prime (p::int) \<Longrightarrow> p > 1"
huffman@31706
  1559
  by (unfold prime_int_def prime_nat_def, auto)
huffman@31706
  1560
huffman@31706
  1561
lemma int_prime_ge_2 [elim]: "prime (p::int) \<Longrightarrow> p >= 2"
huffman@31706
  1562
  by (unfold prime_int_def prime_nat_def, auto)
wenzelm@22367
  1563
huffman@31706
  1564
thm prime_nat_def;
huffman@31706
  1565
thm prime_nat_def [transferred];
huffman@31706
  1566
huffman@31706
  1567
lemma prime_int_altdef: "prime (p::int) = (1 < p \<and> (\<forall>m \<ge> 0. m dvd p \<longrightarrow>
huffman@31706
  1568
    m = 1 \<or> m = p))"
huffman@31706
  1569
  using prime_nat_def [transferred]
huffman@31706
  1570
    apply (case_tac "p >= 0")
huffman@31706
  1571
    by (blast, auto simp add: int_prime_ge_0)
huffman@31706
  1572
huffman@31706
  1573
(* To do: determine primality of any numeral *)
huffman@31706
  1574
huffman@31706
  1575
lemma nat_zero_not_prime [simp]: "~prime (0::nat)"
huffman@31706
  1576
  by (simp add: prime_nat_def)
huffman@31706
  1577
huffman@31706
  1578
lemma int_zero_not_prime [simp]: "~prime (0::int)"
huffman@31706
  1579
  by (simp add: prime_int_def)
huffman@31706
  1580
huffman@31706
  1581
lemma nat_one_not_prime [simp]: "~prime (1::nat)"
huffman@31706
  1582
  by (simp add: prime_nat_def)
chaieb@22027
  1583
huffman@31706
  1584
lemma nat_Suc_0_not_prime [simp]: "~prime (Suc 0)"
huffman@31706
  1585
  by (simp add: prime_nat_def One_nat_def)
huffman@31706
  1586
huffman@31706
  1587
lemma int_one_not_prime [simp]: "~prime (1::int)"
huffman@31706
  1588
  by (simp add: prime_int_def)
huffman@31706
  1589
huffman@31706
  1590
lemma nat_two_is_prime [simp]: "prime (2::nat)"
huffman@31706
  1591
  apply (auto simp add: prime_nat_def)
huffman@31706
  1592
  apply (case_tac m)
huffman@31706
  1593
  apply (auto dest!: dvd_imp_le)
huffman@31706
  1594
  done
chaieb@22027
  1595
huffman@31706
  1596
lemma int_two_is_prime [simp]: "prime (2::int)"
huffman@31706
  1597
  by (rule nat_two_is_prime [transferred direction: nat "op <= (0::int)"])
chaieb@27568
  1598
huffman@31706
  1599
lemma nat_prime_imp_coprime: "prime (p::nat) \<Longrightarrow> \<not> p dvd n \<Longrightarrow> coprime p n"
huffman@31706
  1600
  apply (unfold prime_nat_def)
huffman@31706
  1601
  apply (metis nat_gcd_dvd1 nat_gcd_dvd2)
huffman@31706
  1602
  done
huffman@31706
  1603
huffman@31706
  1604
lemma int_prime_imp_coprime: "prime (p::int) \<Longrightarrow> \<not> p dvd n \<Longrightarrow> coprime p n"
huffman@31706
  1605
  apply (unfold prime_int_altdef)
huffman@31706
  1606
  apply (metis int_gcd_dvd1 int_gcd_dvd2 int_gcd_ge_0)
chaieb@27568
  1607
  done
chaieb@27568
  1608
huffman@31706
  1609
lemma nat_prime_dvd_mult: "prime (p::nat) \<Longrightarrow> p dvd m * n \<Longrightarrow> p dvd m \<or> p dvd n"
huffman@31706
  1610
  by (blast intro: nat_coprime_dvd_mult nat_prime_imp_coprime)
huffman@31706
  1611
huffman@31706
  1612
lemma int_prime_dvd_mult: "prime (p::int) \<Longrightarrow> p dvd m * n \<Longrightarrow> p dvd m \<or> p dvd n"
huffman@31706
  1613
  by (blast intro: int_coprime_dvd_mult int_prime_imp_coprime)
huffman@31706
  1614
huffman@31706
  1615
lemma nat_prime_dvd_mult_eq [simp]: "prime (p::nat) \<Longrightarrow>
huffman@31706
  1616
    p dvd m * n = (p dvd m \<or> p dvd n)"
huffman@31706
  1617
  by (rule iffI, rule nat_prime_dvd_mult, auto)
chaieb@27568
  1618
huffman@31706
  1619
lemma int_prime_dvd_mult_eq [simp]: "prime (p::int) \<Longrightarrow>
huffman@31706
  1620
    p dvd m * n = (p dvd m \<or> p dvd n)"
huffman@31706
  1621
  by (rule iffI, rule int_prime_dvd_mult, auto)
chaieb@27568
  1622
huffman@31706
  1623
lemma nat_not_prime_eq_prod: "(n::nat) > 1 \<Longrightarrow> ~ prime n \<Longrightarrow>
huffman@31706
  1624
    EX m k. n = m * k & 1 < m & m < n & 1 < k & k < n"
huffman@31706
  1625
  unfolding prime_nat_def dvd_def apply auto
huffman@31706
  1626
  apply (subgoal_tac "k > 1")
huffman@31706
  1627
  apply force
huffman@31706
  1628
  apply (subgoal_tac "k ~= 0")
huffman@31706
  1629
  apply force
huffman@31706
  1630
  apply (rule notI)
huffman@31706
  1631
  apply force
huffman@31706
  1632
done
chaieb@27568
  1633
huffman@31706
  1634
(* there's a lot of messing around with signs of products here --
huffman@31706
  1635
   could this be made more automatic? *)
huffman@31706
  1636
lemma int_not_prime_eq_prod: "(n::int) > 1 \<Longrightarrow> ~ prime n \<Longrightarrow>
huffman@31706
  1637
    EX m k. n = m * k & 1 < m & m < n & 1 < k & k < n"
huffman@31706
  1638
  unfolding prime_int_altdef dvd_def
huffman@31706
  1639
  apply auto
huffman@31706
  1640
  apply (rule_tac x = m in exI)
huffman@31706
  1641
  apply (rule_tac x = k in exI)
huffman@31706
  1642
  apply (auto simp add: mult_compare_simps)
huffman@31706
  1643
  apply (subgoal_tac "k > 0")
huffman@31706
  1644
  apply arith
huffman@31706
  1645
  apply (case_tac "k <= 0")
huffman@31706
  1646
  apply (subgoal_tac "m * k <= 0")
huffman@31706
  1647
  apply force
huffman@31706
  1648
  apply (subst zero_compare_simps(8))
huffman@31706
  1649
  apply auto
huffman@31706
  1650
  apply (subgoal_tac "m * k <= 0")
huffman@31706
  1651
  apply force
huffman@31706
  1652
  apply (subst zero_compare_simps(8))
huffman@31706
  1653
  apply auto
huffman@31706
  1654
done
chaieb@27568
  1655
huffman@31706
  1656
lemma nat_prime_dvd_power [rule_format]: "prime (p::nat) -->
huffman@31706
  1657
    n > 0 --> (p dvd x^n --> p dvd x)"
huffman@31706
  1658
  by (induct n rule: nat_induct, auto)
chaieb@27568
  1659
huffman@31706
  1660
lemma int_prime_dvd_power [rule_format]: "prime (p::int) -->
huffman@31706
  1661
    n > 0 --> (p dvd x^n --> p dvd x)"
huffman@31706
  1662
  apply (induct n rule: nat_induct, auto)
huffman@31706
  1663
  apply (frule int_prime_ge_0)
huffman@31706
  1664
  apply auto
huffman@31706
  1665
done
huffman@31706
  1666
huffman@31706
  1667
lemma nat_prime_imp_power_coprime: "prime (p::nat) \<Longrightarrow> ~ p dvd a \<Longrightarrow>
huffman@31706
  1668
    coprime a (p^m)"
huffman@31706
  1669
  apply (rule nat_coprime_exp)
huffman@31706
  1670
  apply (subst nat_gcd_commute)
huffman@31706
  1671
  apply (erule (1) nat_prime_imp_coprime)
huffman@31706
  1672
done
chaieb@27568
  1673
huffman@31706
  1674
lemma int_prime_imp_power_coprime: "prime (p::int) \<Longrightarrow> ~ p dvd a \<Longrightarrow>
huffman@31706
  1675
    coprime a (p^m)"
huffman@31706
  1676
  apply (rule int_coprime_exp)
huffman@31706
  1677
  apply (subst int_gcd_commute)
huffman@31706
  1678
  apply (erule (1) int_prime_imp_coprime)
huffman@31706
  1679
done
chaieb@27568
  1680
huffman@31706
  1681
lemma nat_primes_coprime: "prime (p::nat) \<Longrightarrow> prime q \<Longrightarrow> p \<noteq> q \<Longrightarrow> coprime p q"
huffman@31706
  1682
  apply (rule nat_prime_imp_coprime, assumption)
huffman@31706
  1683
  apply (unfold prime_nat_def, auto)
huffman@31706
  1684
done
chaieb@27568
  1685
huffman@31706
  1686
lemma int_primes_coprime: "prime (p::int) \<Longrightarrow> prime q \<Longrightarrow> p \<noteq> q \<Longrightarrow> coprime p q"
huffman@31706
  1687
  apply (rule int_prime_imp_coprime, assumption)
huffman@31706
  1688
  apply (unfold prime_int_altdef, clarify)
huffman@31706
  1689
  apply (drule_tac x = q in spec)
huffman@31706
  1690
  apply (drule_tac x = p in spec)
huffman@31706
  1691
  apply auto
huffman@31706
  1692
done
chaieb@27568
  1693
huffman@31706
  1694
lemma nat_primes_imp_powers_coprime: "prime (p::nat) \<Longrightarrow> prime q \<Longrightarrow> p ~= q \<Longrightarrow>
huffman@31706
  1695
    coprime (p^m) (q^n)"
huffman@31706
  1696
  by (rule nat_coprime_exp2, rule nat_primes_coprime)
chaieb@27568
  1697
huffman@31706
  1698
lemma int_primes_imp_powers_coprime: "prime (p::int) \<Longrightarrow> prime q \<Longrightarrow> p ~= q \<Longrightarrow>
huffman@31706
  1699
    coprime (p^m) (q^n)"
huffman@31706
  1700
  by (rule int_coprime_exp2, rule int_primes_coprime)
chaieb@27568
  1701
huffman@31706
  1702
lemma nat_prime_factor: "n \<noteq> (1::nat) \<Longrightarrow> \<exists> p. prime p \<and> p dvd n"
huffman@31706
  1703
  apply (induct n rule: nat_less_induct)
huffman@31706
  1704
  apply (case_tac "n = 0")
huffman@31706
  1705
  using nat_two_is_prime apply blast
huffman@31706
  1706
  apply (case_tac "prime n")
huffman@31706
  1707
  apply blast
huffman@31706
  1708
  apply (subgoal_tac "n > 1")
huffman@31706
  1709
  apply (frule (1) nat_not_prime_eq_prod)
huffman@31706
  1710
  apply (auto intro: dvd_mult dvd_mult2)
huffman@31706
  1711
done
chaieb@23244
  1712
huffman@31706
  1713
(* An Isar version:
huffman@31706
  1714
huffman@31706
  1715
lemma nat_prime_factor_b:
huffman@31706
  1716
  fixes n :: nat
huffman@31706
  1717
  assumes "n \<noteq> 1"
huffman@31706
  1718
  shows "\<exists>p. prime p \<and> p dvd n"
nipkow@23983
  1719
huffman@31706
  1720
using `n ~= 1`
huffman@31706
  1721
proof (induct n rule: nat_less_induct)
huffman@31706
  1722
  fix n :: nat
huffman@31706
  1723
  assume "n ~= 1" and
huffman@31706
  1724
    ih: "\<forall>m<n. m \<noteq> 1 \<longrightarrow> (\<exists>p. prime p \<and> p dvd m)"
huffman@31706
  1725
  thus "\<exists>p. prime p \<and> p dvd n"
huffman@31706
  1726
  proof -
huffman@31706
  1727
  {
huffman@31706
  1728
    assume "n = 0"
huffman@31706
  1729
    moreover note nat_two_is_prime
huffman@31706
  1730
    ultimately have ?thesis
huffman@31706
  1731
      by (auto simp del: nat_two_is_prime)
huffman@31706
  1732
  }
huffman@31706
  1733
  moreover
huffman@31706
  1734
  {
huffman@31706
  1735
    assume "prime n"
huffman@31706
  1736
    hence ?thesis by auto
huffman@31706
  1737
  }
huffman@31706
  1738
  moreover
huffman@31706
  1739
  {
huffman@31706
  1740
    assume "n ~= 0" and "~ prime n"
huffman@31706
  1741
    with `n ~= 1` have "n > 1" by auto
huffman@31706
  1742
    with `~ prime n` and nat_not_prime_eq_prod obtain m k where
huffman@31706
  1743
      "n = m * k" and "1 < m" and "m < n" by blast
huffman@31706
  1744
    with ih obtain p where "prime p" and "p dvd m" by blast
huffman@31706
  1745
    with `n = m * k` have ?thesis by auto
huffman@31706
  1746
  }
huffman@31706
  1747
  ultimately show ?thesis by blast
huffman@31706
  1748
  qed
nipkow@23983
  1749
qed
nipkow@23983
  1750
huffman@31706
  1751
*)
huffman@31706
  1752
huffman@31706
  1753
text {* One property of coprimality is easier to prove via prime factors. *}
huffman@31706
  1754
huffman@31706
  1755
lemma nat_prime_divprod_pow:
huffman@31706
  1756
  assumes p: "prime (p::nat)" and ab: "coprime a b" and pab: "p^n dvd a * b"
huffman@31706
  1757
  shows "p^n dvd a \<or> p^n dvd b"
huffman@31706
  1758
proof-
huffman@31706
  1759
  {assume "n = 0 \<or> a = 1 \<or> b = 1" with pab have ?thesis
huffman@31706
  1760
      apply (cases "n=0", simp_all)
huffman@31706
  1761
      apply (cases "a=1", simp_all) done}
huffman@31706
  1762
  moreover
huffman@31706
  1763
  {assume n: "n \<noteq> 0" and a: "a\<noteq>1" and b: "b\<noteq>1"
huffman@31706
  1764
    then obtain m where m: "n = Suc m" by (cases n, auto)
huffman@31706
  1765
    from n have "p dvd p^n" by (intro dvd_power, auto)
huffman@31706
  1766
    also note pab
huffman@31706
  1767
    finally have pab': "p dvd a * b".
huffman@31706
  1768
    from nat_prime_dvd_mult[OF p pab']
huffman@31706
  1769
    have "p dvd a \<or> p dvd b" .
huffman@31706
  1770
    moreover
huffman@31706
  1771
    {assume pa: "p dvd a"
huffman@31706
  1772
      have pnba: "p^n dvd b*a" using pab by (simp add: mult_commute)
huffman@31706
  1773
      from nat_coprime_common_divisor [OF ab, OF pa] p have "\<not> p dvd b" by auto
huffman@31706
  1774
      with p have "coprime b p"
huffman@31706
  1775
        by (subst nat_gcd_commute, intro nat_prime_imp_coprime)
huffman@31706
  1776
      hence pnb: "coprime (p^n) b"
huffman@31706
  1777
        by (subst nat_gcd_commute, rule nat_coprime_exp)
huffman@31706
  1778
      from nat_coprime_divprod[OF pnba pnb] have ?thesis by blast }
huffman@31706
  1779
    moreover
huffman@31706
  1780
    {assume pb: "p dvd b"
huffman@31706
  1781
      have pnba: "p^n dvd b*a" using pab by (simp add: mult_commute)
huffman@31706
  1782
      from nat_coprime_common_divisor [OF ab, of p] pb p have "\<not> p dvd a"
huffman@31706
  1783
        by auto
huffman@31706
  1784
      with p have "coprime a p"
huffman@31706
  1785
        by (subst nat_gcd_commute, intro nat_prime_imp_coprime)
huffman@31706
  1786
      hence pna: "coprime (p^n) a"
huffman@31706
  1787
        by (subst nat_gcd_commute, rule nat_coprime_exp)
huffman@31706
  1788
      from nat_coprime_divprod[OF pab pna] have ?thesis by blast }
huffman@31706
  1789
    ultimately have ?thesis by blast}
huffman@31706
  1790
  ultimately show ?thesis by blast
nipkow@23983
  1791
qed
nipkow@23983
  1792
wenzelm@21256
  1793
end